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Full text of "Vistas in Astronomy - Volume 1"

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ASTRONOMY 



Volume One 

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VISTAS IN ASTRONOMY 



Volume 1 



My subject disperses the galaxies, hut it unites the Earth. 



Sir Arthur Eduington, at the Meeting 

of (he International Astronomical Union, 
Cambridge, Mass., U.S. A., September, 1932, 




7.J.H fc^&zi. 



THESE TWO VOLUMES ARE DEDICATED TO 

F. J. M. STRATTON 

D.S.O., C.B.E., T.D., M.A., IION.LL.D., D.PHIL., F.R.S. 

EMERITUS PROFESSOR OF ASTROPHYSICS 
IN THE UNIVERSITY OF CAMBRIDGE 

BY HIS I-R1ENDS AND ADMIRERS 
EVERYWHERE 



VISTAS IN 
ASTRONOMY 

IN TWO VOLUMES 
EDITED BY 

ARTHUR BEER 



Volume 1 



CO-OPERATION AND ORGANIZATION 

HISTORY AND PHILOSOPHY 

DYNAMICS 

THEORETICAL ASTROPHYSICS 

INSTRUMENTS 

RADIO ASTRONOMY 

SOLAR PHYSICS 



PEKCAMON PRESS 

LONDON & NEW YORK 
1955 



Special Supplement No. 3 to the Journal of Atmospheric and Terrestrial Physics 



Published by Pergumon Press Ltd., 4 <& 5 Fitzroy Square, Loudon. W,\ 
ami PergeiMOH Press Inc., 122 E, 55tb Street, New York 22, N.Y. 

Printed in Great Britain by the Pitman Press, Bath, Somerset 



CONTENTS OF VOLUME 1 

Introduction by ARTHUR Reek page , . . xi 

Hommage respectueux, cordial, et reconnaissaul ;i Monsieur le Prnfcsseur K J. M. Stratton: 

. Plaxet 11)42 XH - "STitATrosiA" hy K. Delporte (Brussels) , 

1. CO-OPERATION AND ORGANIZATION 

International co-operation in astronomy by ,\l. Mixxaeut (Utrecht) 5 

Tlit- International Astronomical Union % P. TiL Oosteuhoek (Ixuden) II 

Tli. International Astronomical Xews Service by Juuk V inter Hansen (Copenhagen) Ifl 
The Committee for the Distribution of Astronomical Literature and the Astronomical 

News letters by Bakt .P. Hiik (Cambridge, Mass.) and V. Koi-iMiAXUKi' (Lille) 22 

Sortie problems of internal ional en-operation in geophysics by J. M. StaoG ( London.) 2ti 

Some educational aspects of astronomy by V. BaBOOAS (Preslmi. Sag.) 25* 

Recollections oF seventy years of scientific: work by I, EVHBSHHD, F.ILS. (F.whmsl, Bug.) 88 

Astronomical recollect ions 6// Max Horn, F.ILS. (Had Pyrmont, Germany) 1 1 

2. HISTORY AND PHILOSOPHY 

The Egyptian " Oceans" hy O. XEr^KRAt-KH (Pj-ovklence, ILL) 47 
The astral religion of anliquily ami the •Thinking Machines" of today by Georojs 

Sahtox (Cambridge, Mass.) ol 

Ptolemy's precession by A. Pannekoek (Wageningen, Netherlands) il(> 

A medieval footnote to rtolcmaie precession by Derek J. Price (Cambridge) 0l> 
The Peking Observatory in A.D. 1280 and the development of the equatorial mounting 

6g Joseph Nkedham. F.H.S. (Cambridge) 87 
The Mercury Horoscope of Marcantonio Miehiel of Venice: a study in Hie history of 

Renaissance astrology and astronomy by Wiu.Y Uartxeh (Frankfurt a.M.) Si 
Observatories and instrument makers in the eighteenth century by D, W. Dkwhikst 

(Cambridge) l3u 

Tin- Radcliffe Observatory by IL Kxox-SllAW (Elgin, South Africa) I I I 

A system of quantitative astmnnmiral obsei vat ions by ('. \V. At.l.KN (Ixuulon) I 19 

Fact and inference in theory anil in observation by IL Hoxin (liondon) L86 

Phi tosophical aspects of cosmology by 1 1 ek hert 1) I ncj i.e ( I xmdon ) 1 02 

Modern cosmology and the theologians by M. |)avii»<>n ( Uidley Kud, ling.) ldfi 

3. DYNAMICS 

Vistas in celestial mechanics by I), II, Sadler (llerstmoneeux) 175 

An introduction to 1h<- eclipse inooti by It, d'E. Atkinson (Greenwich) 180 
The Moon's principal librations in rectangular co-ordinal ch by Srit Harold .Iekkkicys, 

Rlts. (Cambridge) 188 

A general expression for a Ijigrangian Bracket hy W. M, SMART (Glasgow) 194 

The mk- of Craeovians in astronomy by T\ ItAXAcinKWErK (Cracow) 200 

(Secularization of the Three-Body Problem by O. LemaItke (Brussels) 207 

Recent, developments in stellar dynamics by (L L. Camm (Manchester) 116 

"Intrinsic" studies of stellar movements in the Milky Way by Pail BOURGEOIS (Brussels) 22 I 



vi Contents 

JKWJV 

The ASerTect in stellar motions by Makouj h\ Weavkh (Berkeley, Cal.) 228 

On the empirical foundation iA the General TheoRf ol Hi>l;iti v itv 69 EL Fjnxay- 

PHEUNnuell (St. Andrews) 239 



4. THEORETICAL ASTROPHYSICS 

Kraunhofer lines ami the structure of stellar atmospheres by A. U>\sor,n (Kiel) and 

V. Wkidkmann (Brunswick) 2 IS) 

On (Ik; formation ol: condensations in a gaseous nebula hi/ II. Zan.htra (Amsterdam) 266 

The calculation of atomic transition probabilities by It. II. Garstano (London) 208 

Atomic collision processes in astrophysics by II. S. W. Massky, F.B.N. (London) 277 

Knergy production in Stars b& El. EL Sai.I'ETKK (Ithaca., N.Y.J 283 
The effective pressure excited by a gas in turbulent motion by <i. K. Uatuhklok 

(Cam bridge) 290 
"Turbulence," kinetic temperature, and electron temperature in stellar atmospheres 

by J\ L. [{iiATNAOAit ( I U-lhi ). M. KltooK (Cambridge. Mass.), 1). EL Mkskki. (Cambridge. 

Mass.) and It. N. Thomas Boulder, Cot) 290 

Tlu> intensity of emission lines in stellar spectra byl J . Wellmann (Ifamburg-Bcrgedorf) 303 

Dynamo theories of cosmic magnetic lields by T. G. Cowling, F.U.S. (Leeds) 313 

On the interpretation ol stellar magnetic lields by S. K, Huncorn (Cambridge) 323 

Theories of variable stellar magnetic lields by V. 0. A. Ferraro (London) 330 

Theories of interstellar polarisation by Lkvuhett Davis, Jr. (Pasadena, Cal.) 336 
The gravitational instability of an infinite homogeneous medium when a Curiolis 

acceleration is acting by H. Ciiandraskkhar, F.1L.S. (Williams Hay, Wis.) 344 

5. INSTRUMENTS 

Modem developments in telescope optics by E. IL Ijnfoot (Cambridge) 351 

Thermal distortions of telescope mirrors and their correction by Asdre Cotjder (Paris) 372 
On the hiterferomelric measurement of small angular distances by Andre Danjon 

(Paris) 377 

Partially coherent optical lields by K. Woi.f (.Manchester) 385 
On a possible improvement of Michclson's method for the determination of stellar 

diameters in poor visibility by IL FfjRTH (London) and B. FiXLAY-l-'uKiNDUi'it 

(Kt. Andrews) 393 

Astronomical spect rographs: past, present and future by I. 8. Bo wen (Pasadena. Cal.) 400 

Spectroscopy with the eehello by GKOiaiK It. Harrison (Cambridge, Mass.) 405 
Comparison of the efficiency of prism and grating spectrographs by Cu. Fehrenkach 

(Maiseill.-) III 

Optical systems employed in apeetroheliograpiba by P. A. Way'man (lierstmonceux) 422 
The solar installation of thmshik Observatory by IL A. and Mary T. Bruce (Dunsink- 

DubMn) 430 
Quarts crystal clocks by Humphry M. Smith ami (i. B. Wellcate (Abinger Common, 

Bug.) 138 
A rapid method of obtaining positions from wide-angle plates by G. van BrESBROBCK 

(Williams Bay, Wis.) 447 

Nomograms in astronomy by 1 1. Strasst. (Hotm) 151 

The use of electronic calculating machines in a,sLronomy by Peter Naur (Copenhagen) 107 

Atp disturbance in rellectors by W. II. Stkavknson (Cambridge) 173 

Photoelectric devices in astronomy by P. IS. Fklluett (Cambridge) 475 

The development of photoelectric photometry by C. KE. Buffer (Madison, Wis.) 491 
On the application of the photomultiplser to astronomical photometry by It. II. Bardie 

(Flagstaff, Adz.) 498 

On ideal reception of photons and the electron telescope by A. Lallemand (Paris) 505 

Indirect methods of star counting by 11. K. Pitlkk (Hdinburgh) B0U 



Contents vu 

6. RADIO ASTRONOMY vme 

Solar radio eclipse observations by J. 8- Hey (Malvern, Kng.) 881 
The application of mterferometric methods in radio astronomy by M. RYI.R, F.R.S. 

(Cambridge) *8S 
l~irge radio telescopes and their use in radio astronomy by H. Et&HBUKX Brown and 

A. <\ it. Uiveix. F.K.N. (Manchester) 542 
Positional radio-astronomy and the identification of some of tin* radio stars by F. G. 

S&HTH (Cambridge) i:)o0 

Australian work on radio stars % I, G. BoLTon* (.Sydney) 508 

Outbursts of radio noise from the Sun by J. P. Wild (Sydney) S7S 

Radio eeho si udies of meteois 6/y J. G. Davies and A. C\ B. Loveix, F. U.S. (Manchester) S§§ 

The scintillation of radio slam ft;*/ A. IIewimh (Cambridge) 608 

Measures of the 21-cin line emitted by interstellar hydrogen by J. H. Ooiit (Leiden) 807 

7. SOLAR PHYSICS 

Fariy solar research at Mount Wilson by W. S. Adams, For.Meni.R,*. (Pasadena, Cal.) <S19 

Thirty years of solar work at Arcetri fcy Gioroio Aiihtti (Arcetri-Firenze) (>2-i 

Some in< jblems of the international solar survey (j// K . O, K.i mi-en 1 1 i;t i kk ( Freiburg LB, ) m 1 

Physical conditions in the solar photosphere by H. EL Plaskktt. F.K.S. (Oxford) 83? 

Infra-red Fraunhofcr lines and photospheric sit-in tare by R. H.J. Pao.EE (Cambridge) MS 
Temperature at the poles and at the equator of the Sun by A.K.Bas (Ktidaikanal, 

India) and K. L). Aheiyaxkar (Berkeley. Cal.) *&% 

Tlie lineage of the great sunspots by H. W. Newton* ( 1 [ersl monceux | 64J8 

The structure of sunspots by P, A, Swket (London) 678 

The iuagm-1 to (iclds of sunspots by Sffi Edvvauo C. Hullakd. F.R.S. (Teddington. Fng.} (i&5 

On mean latitudinal movements of sunspots by Jaakko TVominen (Helsinki) 601 

The evolution of solar prominences by L. d'AzAMBUJA (Meudon) (5°5 
In vesications of (hues and prominences at the Crimean Astrophysical Observatory by 

A. R- Kevewny (Simeis. Crimea) «*1 

Eclipse observing % Sir John A. Cakhoix (London) ""* 

Recent eclipse work in Japan by YUSTJKK LTagihaka (Tokyo) "08 
The widths of narrow lines in the spectrum of the low clmmiuspheiv. measured ;i< the 

total eclipse of 1952 February 25 by R. O. Redman, F.R.S. (Cambridge) il'S 
The excitation temperature of the solar chromosphere determined from molecular 

spectra by l.». K. BLACKWE3K. (Cambridge) '-<> 

Solai work and plans in Egypt % M, R, Madwak (Cairo) "32 

Spectroscopic studies of the solar corona by O. Hiimisi (Arcetri-Fkeoze) 738 

( 'on majjraphie work at moderate altitude above sea level by V Nov !■: Oil ma n (Stockholm) 717 
Spectroscopic measurements of magnetic fields on the Sun by IT. von KeOber 

(Cambridge) " 51 



CONTENTS OF VOLUME 2 



8. SOLAR-TERRESTRIAL RELATIONS 

Regularities and irregularities in the ionosphere by Sin .Kuwakh Appeetox, l\R.S. 

(Edinburgh) 
The microscopic mechanism for the absorption of radio waves in the ionosphere 

bij J. A, ItA'rriJFFE, F.K.S. (Cambridge) 
Sular Mans by M. A. Ki.m-on f Fdinhurghi 
Af -regions and solar activity by M. WajjomeIEH (Zurich) 
The corpuscular radiation of the Sun by M. J. Smyth (Dunsink- Dublin) 
Cosmic rays and the Sun by P. M. S. Blackktt. F.R.S. (lxindon) 



vim Co nl cuts 

9. GEOPHYSICS 

Non-seasonal changes in the Karths eolation by Nik Harold Npkxckr. Jones, F.R.IS., 
Astronomer Royal (Hei-Htnioneeux) 

Seasonal fluctuations in the rate of rotation o|' the Earth by F. II. van oion Dunokn, 

J. F. Cox 1 and J. van Mi emu km (Brussels) 
The thickness of the continents by Rohkrt Ntmnjoley. F.R.s. (Cambridge) 
Some techniques for locating Hiurulexafeorau from a single observing station by EL T, 

PlBBOT (Can i bridge) 

The abundance of atmospheric carbon dioxide and its isotopes by Leo Coldbeko 
(Ann Arbor, Mich, ) 

The infra-rnd spectrum of the Earth'8 atmosphere by T. W. WossMEIi (G&mhtf&ge) 
The ozone layer; an investigation of (lie 0,6 it ozone band in the telluric spectram % 
M. M. (iiP(H)V ji tul C. [>. Wausfiaw (■( 'amhridgo) 

Atmospheric ozone near tins North I'ole by A. ft. Doii-las, (;. [Ierzbkhw, F.R.S. and 
[>. a Rosa (Ottawa) 

Research in 1 ho upper alini isphere with high altitude Hounding rockets 6y S. P. SrxoEli 
(College Park, Maryland) 

The morphology of geomagnetic storms mid bays: a general review by SYDNEY 

Chafman, P.R.S. (College, Alaska) 
The airglow by Daniel Barrier (Paris) 

10. PLANETARY SYSTEM 

Current trends in minor planets' research by Dirk BimrwER (New Haven, Conn.) 
On the present situation in cuinetary research by M. Beyer (Ilauabui-g-Hergedorf) 
The spectra of the comets by I*. Swings (Liege) 

The Harvai-d photographic meteor programme by Lnr;i (.;•, Jacchia and Fred L. 
Whipple (Cambridge. Mass.) 

On the evolution of meteor .streams by Miroslav Pi„vvec (Prague) 

The zodiacal light by II. 0. van* dk Hulst (I^iden) 

11. STELLAR ASTRONOMY 

Recent advances iii positional astronomy by A. Kopff (Heidelberg) 

On the increase of tin; accuracy of fundamental declinations by F. Schmeidlei: 
(Canberra) 

The distances of the stars: a historical review by J. Jackson, F.R,S. (Ewell. Fng.) 

I'iil v years of trigonometrical parallaxes by A, Hunter and E. O. Martin (G-reetiwich) 

The internal accuracy of tile Vale parallaxes by Giwtav Land (New Haven, Conn.) 

The visual binaries by W. II. van DUN BOS (Johannesburg) 

Planetary companions of stars by Peter van OM Kami 1 (Hwarthmore, Pa.) 

White dwarfs and degenerate stars by William J, U;yten (Minneapolis Minn.) 

The dimensions of Omega Centaur) by K. M. Lindsay (Armagh, X. Ireland) 

On the groups of diffuse emission nebulae by (I. A, Siiajn (Moscow) 

The problem of llagen's Clouds by Pkikdiuch Bkckku and J. Meckers (Bonn) 

Interstellar dust by O. Alter (Prague) 

Interstellar polarization by W. A, Hilt nek (Williams Hay, Wis.) 

12. PHOTOMETRY 

Monochromatic mat-nil odes by K.v.n.B. Wodllev, l*\R.S. (Canberra) 

The Cape magnitudes and their relationship to the international system by R. H. Stoy 
(Cape) 

The zero-point in stellar photometry by ft. Kyijka (Wroclaw; 

Photographic photometry of the Ityades by O. Heckmann (llamburg-Bergedorf) 



Contents 

Note on the diameters of pbefcOTisual star images 6g BjhaH SBBIEZSPIWWQ fTeBese, 

Denmark ) 
The galactic cluster Bd 2!) (NGC 801S) '>;/ VV. \V. Mi>i«mn' ami M. L, Nahki* (Williams 

Bay, Wis.) 
A contribution bo the problem of the classification of variable stars &$f UviO Gkatton 

(Eva Peron, Argentine) 
Population M - Cepheids of the Galaxy by Cecilia I'aynk-Gaihisc jikin (Cambridge, 

Mass.) 
Spectrophotometric observations of Cepheids at Areotri by Maiu;iiaihta Hack (Arcetri- 

Ffrenze) 
The problem of KK Lyrae stars with several periods by L. Dsran (Budapest) 
Photographic Ught-OUrve Of Eta Carinae />]/ D. JT. K- (POOHSHIIX, S.J. (Castelgandolfo, 

Vatican City) 
Eclipsing variables— a survey of the field by Fn\sw Huahshaw Wood (Upper Darby, 

Pa.)" 
Eclipsing variables in the Magellanic Clouds by 1 1 bhb* n. -aais Bussaii, Por.MeuuR.S. 

(Princeton, N.J. ) 
On finding the apBidal period of an eclipsing binary by 3. DS Koht, S..I . (Castelgando I fu, 

Vatican City) 
Photometric effects of gaseous envelopes in close eclipsing binaries by Mu-haisl W. 

Oven ijhn (< dusgow) 
Statistics and recent results on eclipsing binaries by Makio (i. EfoAG&STO&O (Catania) 

13. SPECTROSCOPY 
Atomic spectra— their role in astrophysics by CHARLOTTE E. MoOtUB (Washington, D.C.) 
Methods in stellar spectroscopy f>;/ Theodore Digram, Jr. (Rochester, N.V.J 
The ijuantitativc chemical analysis of early- type stars by Lawkbnce II. Aller (Ann 

Arbor. Mich.) 
The technique and possibilities of astronomical spectrophotometry by Jesse L. 

(JKEENSTEIN ( l'a.sadelia. CaL ) 

The continuous spectrum by W, M, H. Greaves, F.R.S. (Edinburgh) 

The problem of stellar temperatures by H. Xienle (Heidelberg) 

New spectral classifications wit li two or three parameter* by I). Ckai.onuk (Paris) 

Two-dimensional spectral classification of stars through photoelectric photometry with 

interference filters by Husut Stuomorex (Williams Hay, Wis.) 
Luminosities of the H stars from spectroscopic measurements by l(. M. I'ktkie 

(Victoria, B.C.) 
Luminosity classification of stars of late spectral type from spectra of moderate dis- 
persion by John F. M k a aD (Toronto) 
Objective-prism speetBtt Of relatively cool stars in the near infra-red by .1. J. Nassau 
(Cleveland, Ohio) 

14. SPECTRAL PECULIARITIES AND NOVAE 

Spectroscopic phenomena in the system of Algol by Otto Stucve, Eor.Mem.lCS. 

(Berkeley, GftL) 
Stare with expanding atmospheres by Paul W. Merrill (Pasadena. Cal.) 
.Shells around P OygDJ .stars by A, O. Thackeray (Pretoria) 
The masses of spectroscopic binaries by Arthik Hker (Cambridge] 
Evidence for and against binary hypotheses of the blue-red erupting stars by Maktin 

Johnsu X ( B i nu i n ghara ) 
Observational results on variable stars with composite spectra by Tchkno Mao-Lin' and 

M. BLOCH (Lyon) 
The peculiar A-stai*s by Arhix J- Deutsch (Pasadena, CaL) 
The carbon stars— an as! ronomical enigma by William I*. Hidi;lman (Mount Hamilton, 

Cal.) 



* Contents 

(Vlestiat emission line objects erf early typo by V. s. Uiiai.s, KK.s.. and .J. V. 

RoTTKNHKitu (Ottawa) 
Kmission lines and stellar atmospheres hy U. |{. Hrnmi>t;K atui K. M. BURBIDGE 

(Cambridge) 

Identification of emission lines in (he spectra of Wulf-Kavcl .stars h,/ Bkvot tiDXtiS 
(Lund) 

Unuma Cassiopeia* fry D. L, Howards (Sidmouth, Eng.) 
The spectra of novae by Dean K. MrfiAUaHLIN (Ann Arbor, Mich.) 
The mechanism of ejection of matter from novae by H. H. IfuSKBL (ttiiueis, Crimea) 
On i |k. nature of physical pmresses underlying the nova phenomenon b>, ZlH0\FK 
Kcil'Al. (Manchester) 

Novae observed by Keiujki (Japosc hkin (Cambridge, Mass.) 

On the relics of super-novae by \V. If. Kamsky (M;uicrVsierj 

15. GALAXIES 

An analysis of the Milky Way % Wii.hklm ISkckku (Basle] 

Milky Way st i ■uetniv in .Sagittarius and Carina; a study in contrasts by Hakt J. BOK 
(Cambridge, Mass.) 

Inrra-rt'd photography in the Milky Way. particularly in Sagittarius b>f Jkan Dikay 
(Lynn). .Tiiski'ji-IIknki Hkjav (Lyon), and Pjkhhk Bkhthiku (St. Michel, B.-Alpea) 
The great nebulae of the Southern sky by Daviij S. Kvans (Pretoria) 
Oa lactic and extragalactie. studies at the Harvard Observatory by H,\iu,mv Shai'I.ky 
(Cambridge, Mass.) 

The distribution of the galaxies by C. [>. Sham-: (Mount Hamilton, Cal.J 

The distribution of bright, galaxies and the Loral Supergalaxv h lf (fftjiAKD l>K 
\' vn (ifi,i-;i s;- (< 'anhemi I 

On metagalactic distance-indicators by Knit Lundmauk (Lund) 

Bed-shifts in the spectra of extragalactie nebulae by Milton- L. HuhajSON (I'asadena 

Oat.) 

16. COSMOGONY AND COSMOLOGY 

Tbe origin of the satellites arid the Ti-ojans by OKHAttn 1'. KriPi-:i< (Williams Hay, Wis.) 
The origin and significance of the Moons surface hr/ Hai«ji,1) C. 1'uky Kur Mem R S 
(Chicago, III.) 

The frequency of meteorite falls throughout Hie ages by l-\ A. I'axith. KILS, (Mainz) 
The essential nature of accretion by H. A. Lyttlkton, F.R.K. (Cambridge) 
On the production of groups of U- and B -stars by accretion by W. 11, McChka, F.R.S. 
(London) 

The elTeel of galactic rotation on accretion hy main sequence stars bit V. Hoylk 
(Cambridge) 

On the origin of double stars hy V. A. AMHAitrsoiiAN (Bjurakan. Armenia) 

On the evolution of stellar systems by Bkktm. Lixdblad (.Stockholm) 

(...sinology by'W Goi.n (Ileiwtmoneeux) 

The physics of the expand ittg universe hy ii. Oamow (Washington, D.O.) 



N'amk Indkx 
siujkct Index 



INTRODUCTION 

One of the results of F. J. M. Stratton's imaginative understanding of human 
affairs has been to bring about harmonious co-operation between the very different 
personalities populating the sphere of astronomical studies. Inspired by his magnifi- 
cent example, I have tried in this book to bring together and to present in an 
organized manner the ideas that form the structural units of present-day astronomy. 

Everyone familiar with our subject knows that it is a field in which controversy 
cannot be avoided ; indeed, at the frontiers of expanding knowledge, controversy is 
most desirable. Contradictory views will therefore sometimes be found here side by 
side with one another. 

It can be questioned whether a synthesis such as was achieved in the seventeenth 
century through the work of Galileo, Kepler, Newton, and others, is attainable in 
the twentieth century. But astronomical science can only advance, as it did then, 
by boldly combining observational and experimental work with the tireless quest 
for theoretical unification. As long as we realize that theoretical interpretations can 
only be held valid until further notice, that is until incontrovertible evidence gained 
by observation compels us to modify them, there can be no objection to the con- 
struction of hypotheses. Newton's own motto "hypotheses non jingo" was, in a 
sense, disregarded by Newton himself: he rejected hypotheses only where they 
violated his own "regula philosophandi" , that is to say, his principle of their strict 
parsimony. In terms of present-day methodology, we reject hypotheses as scientifi- 
cally meaningless if they are incapable even of indirect test ; and we reject them as 
superfluous or as implausible if they are too complex and artificial to conform with 
well-established canons of inductive probability. But freedom of scientific theorizing 
must be preserved wherever the conditions of meaningfulness and of economy 
appear to be satisfied. 

Moreover, we cannot appraise the validity of theories or hypotheses unless a 

great deal of observational material has been accumulated, scrutinized, checked, and 

counter-checked. Astronomical and astrophysical hypotheses, involving as they do 

tremendous extrapolations, must remain controversial for a long time. It is my hope 

that the rich material here assembled, both of observation and of theory, will promote 

the synthesis of our knowledge, leading us per noctem ad lucem, no matter how long 

and tortuous the path may be. 

* * * 

When I first approached our colleagues all over the world to sound them about a 
"Stratton Volume", any possible doubts as to the feasibility of the project were 
quickly dispelled by their warm and encouraging response. 

In my invitation to contributors, I emphasized that the purpose of the book 
was to present original articles and reviews, reflecting fully and critically the present 
situation and recent advances in astronomy and related fields. The authors were also 
asked to frame their contributions with a view not only to their value for the expert, 
but also to their use by less specialized readers wishing to gain a general idea of the 
latest developments. The average length of an article was to be two to three thousand 
words, of which the first few hundred words were to be a statement of the problem 



xu Introduction 

with a brief summary of previous work, while the concluding paragraphs were to be 
devoted to a survey of unsolved questions and possible "vistas" of future research. 
In this way, each article was designed to be a self-contained unit, combining the 
author's own researches with his views on the historical roots of his problem and on 
its future beyond the limits of to-day. 

We thus intended to develop something new, neither a "Handbuch"^ nor a 
"Festschrift" — but a generous and thorough cross-section through the whole of 
contemporary astronomy and the allied sciences, always emphasizing the lines along 
which research is particularly active at the moment, its new techniques and methods, 
and their interaction with theoretical developments. Arrangements were made to 
ensure that every article would add a new and significant facet to the whole. 

It is hoped that this structure of the book may make its contents more accessible 
to those whose interests lie in adjacent fields — such as physics, mathematics, and 
geophysics — than is normally possible with more specialized astronomical texts. 



The original set of collaborators came from the wide range of Professor Stratton's 
immediate colleagues, pupils, and friends. The momentum of the work carried it 
forward until ultimately 215 authors joined to produce the 192 contributions: 179 
of them are astronomers; 36 are physicists, geophysicists, mathematicians, and 
historians. They cover the span between the ages of twenty-four and ninety-one, 
and the more significant one between twenty-six nations. 

The aim was to include one of the leading representatives in every special field. 
Overlapping has been avoided, except where several authors were deliberately 
invited to write on the same topic (a fact which was known to the editor only) — 
approaching it from different angles, with different points of view, and with divergent 
results — thus illuminating the situation with the spotlights of opposing new ideas. 

Some of those who were originally approached found themselves unable to contri- 
bute because of the time-limit necessarily imposed. Happily, they were only few in 
number, and I should like to record my gratitude to them for their good wishes for 
the success of the venture. Their warm words were among the many sources of 
inspiring encouragement extended to me. It is deeply regretted that death has 
prevented the contributions by Bernard Lyot, Edwin Hubble, and Walter 
Grotrian from being included in these volumes, whilst it is sad that the one 
by Thaddatjs Banachiewicz, who died in November, 1954, must remain his 
last work. 

Practically every country in the world with an active astronomical life has contri- 
buted and many languages were involved. For the sake of uniformity it was decided 
to publish everything in English. The editor had, therefore, to provide some 600 
printed pages of translation ; in most cases the whole text had to be turned into 
English; in others the author's provisional English version was revised. The 
problem facing every translator is the search for a suitable compromise between the 
demands of style and those of accuracy. "Les traductions sont comme les femmes: 
quand dies sont belles, elles ne sont pas fideles; et quand elles sont fideles, elles ne sont 
pas belles. " On the whole, I have tried to retain the flavour of the original language 

* "While the 'Handbuch' is obviously important to every astronomer, it is clear that such enormous editions cannot appear 
frequently. Other means must, therefore, be devised to meet the need of scientific summaries. It is to be hoped that a suitable 
organization will undertake to supplement the 'Handbuch' by publishing from time to time relatively short summarizing 
articles containing reviews of new developments in various fields". — Otto Struve, reviewing the first six volumes of the 
Handbuch der Astrophysik in the Astrophysical Journal, vol. 80, p. 73, 1934. 



Introduction xiii 

rather than to produce a uniform and perhaps colourless version. Thus, while every 
effort was made to ensure a smooth flow of language, equal care was taken not to 
impress an impersonal style on the work of colleagues from abroad, each of whom 
wrote in the lively idiom of his own tongue. The final responsibility for the result 

is mine. 

* * * 

I cannot resist the comment that in attempting this cross-section of modern 
astronomy I have been vividly confronted with a cross-section of contemporary 
astronomers in all their delightfully individual ways. The suggested length of the 
papers was some 2500 words : the actual mean length turned out to be 4780 words, 
and the standard deviation 2920 ; the minimum was 360 words, the maximum 28,000. 
The average delay in submitting the manuscripts was 175 days, and the standard 
deviation 95; the extreme values were — 70 days and + 680 days. Many of the 
longer delays were, of course, due to circumstances beyond the authors' control, and 
the overall impression left in my mind is that of the sincere effort made by the 
contributors to keep to the schedule. 

There was also quite a lively exchange of letters. Often these were concerned with 
entirely necessary points. They added to the labour, but did not detract from the 
enjoyment of editing. A total of some 6000 letters proved sufficient to ensure smooth 
relations between contributors, editor, and publisher. 

The suggestion that each article should conclude with a "vista" has been inter- 
preted in many ways, and not everywhere is such an account presented in words. 
One of our most distinguished astrophysicists, for instance, concluded his article with 
a capital Q, and another one with the word "false". I was sometimes surprised by 
the range of topics which made their appearance ; on p. 89, for example, there is a 
discussion of the variability of the human gestation period, and reference 14 on 
p. 490 concerns a decerebrated albino rabbit. On the other hand, it was decided to 
reduce discussion of some familiar topics where recent accounts or new textbooks 
have been published or are in preparation. This is, for instance, the case with certain 
theoretical problems of stellar atmospheres and stellar interiors, the planetary 
system, aurorae, and cosmical aerodynamics. 

Every effort has been made to keep the articles up to date by additional work and 
amendments during the growth of the book. However, the general framework of 
each article had to be fixed before printing could begin, and the mean epoch of final 
revision is 1954-5. 

While the simultaneous publication, in a fixed order, of 192 contributions from 
twenty-six different countries has necessarily been slower than if they had been dealt 
with individually like papers in a scientific journal, it is hoped that none of the 
articles is so ephemeral that its interest will have diminished during the time of 
printing. The generous inclusion by authors of accounts of their latest researches has 
involved some sacrifices in the speed of announcement, but has added a refreshing 
note to the impression which is given of the contemporary astronomer's working day. 

No doubt a number of mistakes and misprints have escaped scrutiny; but it is 
hoped that serious ones have been avoided. In one or two cases manuscripts left on 
my desk were found to have mysteriously grown unsolicited additions, not always of a 
strictly scientific nature; it was discovered just in time that a medieval drawing 
had been provided with the caption "The Editor, half-size". I should be grateful to 
receive notice of any omissions and errors found by readers. 



xiv Introduction 

The system of abbreviations of references is that used in Science Abstracts (of the 
Institution of Electrical Engineers, London; in association with the American 
Physical Society, etc.), and I am much indebted to Dr. B. M. Crowther who kindly 
gave permission to adopt this system and to use it throughout the whole book. The 
request to the authors that "references should be given generously" has resulted in 
the accumulation of a thorough bibliography covering a very wide range of astronomy 
and allied fields, which it is hoped may be not the least valuable part of the work. 



"I owe debts of gratitude to many more people than can conveniently be named, 
people of all degrees and many nationalities. He who befriends a traveller is not 
easily forgotten, and I am very grateful indeed to everyone who helped me on a long 
journey". These words of Peter Fleming in his Travels in Tartary very well suit 
the case of the editor of Vistas. 

I should like to mention some of the many friends and colleagues who have 
always been ready to assist me. 

First of all, I wish to say how sincerely grateful I am to Professor R. 0. Redman, 
without whose help and understanding forbearance the task of editing this book 
could not have been accomplished. 

I am most obliged also to Sir Edward V. Appleton, Dr. G. Bing, Prof. T. G. 
Cowling, Dr. D. W. Dewhirst, Prof. H. Feigl, Dr. R. M. Goody, Prof. W. 
Hartner, Prof. W. H. McCrea, Prof. M. Minnaert, Dr. J. Needham, Dr. D. H. 
Sadler, Dr. M. J. Smyth, and Dr. A. D. Thackeray, for their active interest and 
support. Nor shall I ever forget the stimulating friendship and devoted help which I 
received from Dr. Peter Fellgett and Dr. Bernard Pagel. 

My wife and my daughter Nova (whose name was proposed by Professor Stratton 
in one of those happy nights when he and I joined in the observations of Nova 
Herculis) have helped and encouraged me in many ways. 

The Mount Wilson and Palomar Observatories have been very helpful: Dr. I. S. 
Bowen and Dr. M. L. Humason provided some of the latest photographs used in the 
preparation of the book. I am also much obliged to Dr. Walter Baade, who placed 
his original photograph of the radio source in Cassiopeia at the disposal of Vistas 
(p. 566), as well as his new and previously unpublished record of the peculiar galaxy 
NGC 2623, which is shown in the frontispiece to Volume 2, and is now known to be 
a radio source in Cancer. In the same frontispiece is shown a photograph of the 
interesting double galaxy in Pisces, also believed to be a radio source. For this last 
photograph I am much indebted to Dr. Fritz Zwicky, and for that of NGC 7293, 
on p. 259, to Dr. R. Minkowski. 

Thanks are due to the Master and Fellows of Gonville and Caius College, Cambridge, 
for permission to reproduce their fine portrait of Professor Stratton, painted by 
Sir Oswald Birley, which forms the frontispiece of this volume ; and particularly 
to the Master, Sir James Chadwick, for his kindness in making the necessary 
arrangements. 

I gratefully acknowledge the unfailing helpfulness and disinterested care which the 
publishers devoted to the production and publication of this book, especially that 
of Dr. P. Rosbaud, Mr. E. J. Buckley, and Mr. J. G. Mason, who always gave 
every possible consideration to my suggestions. 

I should really have mentioned some 200 more names: I can only say how sad 



Introduction xv 

I feel that the ways of editor and collaborators must now part again, after such a 
pleasant time together. 

* * * 

It can rarely, if ever, have happened before that so many leading scholars in a single 
discipline have co-operated in one book on such a wide international scale. That this 
was possible is a fitting tribute to the international spirit which has permeated 
Professor Stratton's life-work in our science.* I am glad to be able to include here 
the following passages from a letter recently received from Dr. Walter S. Adams, 
of Mount Wilson Observatory : 

Of the many important contributions to the fields of solar and stellar spectroscopy which 
Professor Stratton has made, I am inclined to rate among the highest his series of investigations 
of the spectra of novae which began in 1912 with Nova Geminorum. He brought to this great 
insight, and I think of him as a pioneer in the modern research which has given us so much infor- 
mation regarding the physical state and constitution of these extraordinary stars. 

I also want to mention as probably the finest relatively short treatise in its field his Astro- 
nomical Physics, first published in 1925. The choice of material, the conciseness with which it is 
handled, and the simple and admirable style of writing still make this a reference book for 
astronomers and laymen in spite of the quarter century since it appeared. 

Stratton was elected General Secretary of the International Astronomical Union at the 
Cambridge Meeting in 1925, succeeding Professor Alfred Fowler, and continued to serve 
until 1938. During this time he prepared and edited the reports of the I.A.TJ. Meetings at 
Leiden, Cambridge in Massachusetts, and Paris. An enormous amount of labour was involved 
m these volumes, and Stratton's pre-eminent qualities of energy, tact, and friendliness were 
tested to the full in dealing with contributors scattered all over the face of the Earth many of 
them with highly individualistic ideas of what should go into a report. In this respect Stratton 
had much the same experience as Professor Arthur Schuster, who as President of the Inter- 
national Union for Co-operation in Solar Research (the predecessor of the I.A.U.) was moved 
to say at the close of the Mount Wilson Meeting in 1910 : "Gentlemen, the reports of the sections 
and committees will be printed in the Transactions of the Union. I wish to caU your attention 
to the fact, however, that in the past some of these reports as submitted have borne little or no 
resemblance to what was said or done at the meeting". 

The contribution which Stratton made to the efficiency and smoothness of operation at the 
meetings cannot be overestimated. A large group of international scientists, each an individualist 
in his own right, often presents a serious problem of organization, but Stratton handled it with 
remarkable skill and diplomacy. He could make allowances for questions of national prestige 
which occasionally arose even in a scientific meeting, and his linguistic ability was of immense 
value on these and similar occasions. I always remember the delightful atmosphere of good 
fellowship which accompanied the close of the Stockholm Meeting in 1938 after Stratton had 
presented to Sweden the greetings and thanks of some twenty different nations, each in its 
native tongue. 

I have many personal memories of Stratton, associated with all sorts of occasions His 
orbit was quite unpredictable as was illustrated by a telegram received on Mount Wilson during 
one of the years of the Second World War. It was from San Francisco and read- "Arriving 
Los Angeles 9.30, (signed) Stratton". Since I had no reason to think that Stratton was less 
than 5000 miles distant at the time, one can imagine the thrill this message produced. We met 

* Stratton, Frederick John Marrian; D.S.O., 1917- OBE 1929 • T n 1924- f r ^ iq^7- ivr a /r. * v, x TT 



xvi Introduction 

at many places on the European Continent, in his rooms at Caius College in Cambridge, in 
London, and in California. 

Two out of many such occasions remain especially in my memory. The first followed the 
organization meeting of the I.A.U. at Brussels in the Summer of 1919. Stratton took a small 
group of us to the battlefields of Ypres and under his pilotage we went through the dug-outs in 
which he had lived so long and given all he had to the cause of Britain in this desperate struggle. 
We climbed Mount Kemmel under a grey sky, and among the trenches and the broken guns and 
refuse of war Stratton outlined to us the course of the great battles which marked a turning- 
point of history. 

A second memory is of the I.A.U. Meeting in Paris in 1935. It was Bastille Day in France, and 
after dinner on the Eiffel Tower, Stratton, who is a true cosmopolitan and knows Paris as well 
as he does London, took a few of us to see the dancing in the streets. He knew just where to go, 
and the Americans in the party received one of the thrills of their lives and a knowledge of the 
French people they could have learned in no other way. . . . 

Another sidelight is contained in the following account, taken from a letter from 
Dr. A. D. Thackeray, of the Radcliffe Observatory; it refers to the Cambridge 
expedition to Japan for the 1936 total solar eclipse (see also pp. 708-10): 

On the evening of June 19th, 1936, after the bitter experience of a cloudy eclipse, the following 
telegram, which we were told had been transmitted in Japanese characters, was placed in our 
hands : 

NOBANIYA EPUSHIRON SEFUEISADOMA CUNICHIYUDO. 
Stratton cheerfully sat down to disentangle the meaning with members of the expedition, and 
in less than half an hour turned out the message as "Nova near epsilon Cephei third magnitude". 
(There was no means of indicating beginnings or ends of words in the message.) On this occasion 
the press reported that we took half an hour to find the nova, without reference to the decoding 
problem. . . . 



I should like to add a personal footnote to my Introduction. Circumstances have 
given me the opportunity of long-continued contact with Professor Stratton 's 
radiant and inspiring personality; it is an experience which I cherish more deeply 
than I can readily express and which I consider one of the greatest gifts of my life. 
If the work on these volumes, dedicated to him, involved many difficulties and 
complications, I have been more than rewarded by the spirit in which a group of 
brilliant and creative minds has co-operated. We should regard the international 
band of 200 astronomers working together to produce this book in honour of a great 
scholar - as an example of what can and should be done in a wider sphere. 

If it be possible for this book to strengthen the appreciation of human efforts 
unhampered by national barriers, and to serve as a symbol of what good will can 
achieve in our troubled world, then, indeed, it will have fulfilled all our hopes. 

The Observatories, Arthur Beer. 

University of Cambridge. 
March, 1955. 



Hommage Respectueux, Cordial, et Reconnaissant a Monsieur le 
Professeur F. J. M, Stratton 

E. Delporte 

Diret-tcur Honorftin 1 de 1'Observatoite Royal de Bclgtque u Ut'i'Io, Associate R,A,S. 



Jl desire que tna petite planete 1042 XB qui a recu le nmncro 1560 parte ]e nom de 
"Stnittoniii" (mi riionneur du present J ubilaire. 

Des voix autorisees retraceront la. earn ere si toufr'uc, si vivante, si profondement 
marquee dans la vie astronomique internationale taut ]>ar ses travaux personnels 
que par la part active qu'il a prise dans lauaissanec et la vie de 1'Union Astronomique 
Internationale. 

.Pour la liolgique, le Proiesseur b\ J. M. Stratton int. Tun de ses aauveteurs lors 
de la premiere guerre mondialc. II tut le representant combien autorise de la Royal 
Astronomical Society aux ceremonies du Centenaire de Conservatoire Royal de 
Belgique en 1935, 

Personnel lenient, je tiens a rappeler la part grande qu'il assuma lors fie la publica- 
tion par 1' Union Astronomique Interna tiou alt; do ma ''Delimitation Seientin'que des 
Constellations" et surtout la generalise collaboration qu'il daigna m'acconler pour 
la publication de V Atlas Celeste paru sous les uietnes auspices de I'U.A.L, avec Jes 
uonvelles li mites. 

Je m'honore d'avoir pu etre considere par M. le Professeur F. J. M. Stratton 
uomine un ami. 



+28.408 



^8.*** 



Fig. t. Plant*. STRATTON IA 

I 



2 Hommage Respectueux, Cordial et Reconnaissant a Monsieur le Professeur F. J. M. Stratton 

La petite planete 1942 XB = 1560 a les elements suivants: 

Equinoxe 1950,0 
m g 1943 M oo Q, i <p /u a 

14,4 11,0 III 17 52°234 92?208 290°172 6°283 12°236 805;'959 2,6861 
d'apres l'ephemeride des petites planetes de Leningrad pour 1953 apres application 
des perturbations speciales. 

Les premiers elements en ont ete calcules par M. Museist au moyen des positions 
du 3 decembre 1942, 4 Janvier 1943, et 1 fevrier 1943. La numerotation 1560 a 
ete annoncee dans la circulaire 2468 de l'Astronomisches Rechen-Institut. La 
planete a ete reobservee aux oppositions de 1946 et 1950. 

J'ai trouve la petite planete sur la plaque No. 1617 prise le 3 decembre 1942 au 
soir, au double astrographe Zeiss par la methode de Metcalf, pose de l h 50 m a 
2 h 43 m , par legere brume. La dite methode a ete employee pour la recherche d'une 
autre petite planete d'eclat faible. Confirmee par la plaque 1620 (astro Agfa) posee 
le 10 decembre au soir, par la meme methode de Metcalf, avec une duree de 56 
minutes, la planete fut suivie par les plaques 1621 (10 decembre soir), 1623 de la 
meme nuit (ciel peu clair), 1627 du 27 decembre au soir, 1628 du 4 Janvier 1943 au 
soir, 1630 du 5 Janvier au soir, 1633 du 8 Janvier au soir, 1638 du 12 Janvier, 1639 
du 13 Janvier et 1640 du 1 fevrier 1943. 

L'eclat assez faible de l'astre et surtout le ciel peu propice pendant toute la periode 
a oblige l'observateur a se servir constamment de la methode Metcalf, concentrant 
la lumiere renvoyee par l'astre en un point, alors que les etoiles tracaient des traits sur 
la plaque. La copie d'une partie de la plaque de decouverte reproduite ici, accuse la 
difficulte de la recherche; les 3 etoiles ayant servi a la mesure de la position sont 
entourees d'un cercle, tandis que l'image de la planete se trouve entre 3 points. 



SECTION 1 



CO-OPERATION 

AND 
ORGANIZATION 



"That tho' a Man were admitted into Heaven to view 
the wonderful Fabrick of the World, and the Beauty 
of the Stars, yet what would otherwise be Rapture 
and Extasie, would be but a melancholy Amazement if 
he had not a Friend to communicate it to." 

Attr. to Archytas by Christianus Huygkns, 
The Celestial Worlds Discovered, Book 1, pt. 4, 1722. 



International Co-operation in Astronomy 

M. MlNNAERT 

Sterrewacht, Utrecht, Netherlands 
Summary 

Scientific co-operation between the nations is found already in Antiquity and the Middle Ages and has 
proved a strong stimulus to the development of astronomy. Different forms of modern international 
co-operation in astronomy may be distinguished: (1) co-ordinated observations at widely separated 
stations; (2) collective achievement of a great amount of work; (3) creation of international centres; 
(4) unification of notations and terminology. The increasing need for co-operation in astronomy was the 
reason for the constitution of international bodies, among which the I.A.TJ. acquired the greatest impor- 
tance; the history of the Union shows that scientific co-operation must be kept outside political implica- 
tions. International meetings, colloquia, travels, and exchanges should be encouraged. The introduction 
of an auxiliary international language would be highly desirable. International co-operation is a necessary 
complement to the national development of science. 



1. Early Forms of Co-operation 

From the earliest periods of civilization, contact between the nations has stimulated 
the development of science. In Antiquity, it was especially the interaction between 
the Oriental and the Greek world which had important consequences. Thales and 
Pythagoras, later Plato and Eudoxos, travelled to Egypt and were initiated into 
the wisdom of the priests. In 280 a.d. the Babylonian Berossos taught astronomy to 
Greek scientists at Kos. The transmission of astronomical learning from the Greeks 
to the Romans and the Arabs, later from the Arabs to Western Europe, was made 
possible by numerous laborious translations of astronomical textbooks, and it saved 
the continuity of the growth of astronomy. 

At the end of the Middle Ages, the national groups were taking shape, and in each 
country centres of science developed with their particular schools of learning, influ- 
enced by exchanges and scientific relations. So, as early as in the thirteenth century, 
we find John Holywood, of York, as a professor of astronomy at the Paris Univer- 
sity. And when the Renaissance stimulated all sciences and arts to a wonderful 
flowering, it was the international contact which diffused the new concepts, brought 
fresh ideas, and new impulses to a rejuvenating world. At most universities, students 
of many nations assembled, easily following the teaching in Latin. It became a 
custom that scientists in their youth should visit foreign countries in order to com- 
plete their education. Purbach travelled to France and Italy; Regiomontanus 
to Vienna, Rome and Hungary; Nicolaus of Cusa to the Netherlands, Germany, 
and Italy; Copernicus to Bologna, Rome, and Padua; Rheticus from N. Italy 
to Copernicus at Frauenburg; the young Tycho to Germany; Simon Marius from 
Francony to Tycho and to Italy. Brilliant scientists were called as professors to 
foreign universities, in the same spirit of internationalism which is our pride nowadays. 
The printed book conveyed knowledge from one country to another and became the 
main instrument of international scientific co-operation : no other is so effective, so 
easy, so permanent. 

However, the diffusion of new discoveries by books takes too much time and is not 
efficient for quick scientific intercourse. In hundreds of years, the real contact be- 
tween actively working scientists was entertained by direct correspondence. The 
letters of Huygens, for example, comprise nearly one-half of his whole scientific 

5 



6 International co-operation in astronomy 

production and have been published, together with the replies, in ten quarto volumes 
of about 600 pages each ; they are addressed to scientists all over Europe. It was only 
in 1679 that the Connaissance des Temps appeared as the first important astronomical 
year book. From 1800 to 1813 the first astronomical periodical was edited by Baron 
von Zach, the Monatliche Korrespondenz, followed after a short interruption by the 
Zeitschrift fur Astronomie (1816-18) and the Correspondance Astronomique (1818-26) ; 
astronomers of several nations contributed papers and the editors endeavoured to 
give information about the development of astronomy all over the world. The 
important astronomical reviews which have originated since then have each kept their 
national character more or less, though papers of foreign colleagues are in general 
welcomed. This international contact has been considerably increased by the 
publication of regular observatory annals which are freely distributed to all foreign 
observatories, according to a system almost unique among scientists and testifying 
of the generosity of the great institutes in favour of the minor ones. 

2. Modern Co-operative Enterprises 

The aspects of international co-operation thus far described do not yet include the 
form which is considered nowadays as the most typical : an organization, consciously 
planned for team work. In 1736, when the French Academy of Sciences wished to 
compare an arc of the meridian at different latitudes, two expeditions were prepared, 
one to Lapony, the other to Ecuador ; nobody considered the possibility of asking 
foreign countries to make independent observations which could be compared after- 
wards. Similarly, in 1671, the parallax of Mars was determined from the results of 
two French astronomers, Richer at Cayenne and Cassini at Paris. 

Probably the first really co-operative international enterprise in astronomy was 
undertaken in 1761 and 1769, when the transit of Venus was observed by numerous 
expeditions with a view to determining the solar parallax. The preliminary calcula- 
tions were derived from an intercomparison of observations from a few distant 
stations; the final discussion, published by Encke in 1822 and 1824, was based on 
the individual results of all the stations. This then may be called the first type of 
worldwide astronomical co-operation: when observations are needed, made at 
distant stations, it is quite natural to combine the efforts of observatories of different 
nationalities. Such combined efforts for the determination of the solar parallax were 
repeatedly made later. For the Venus transits of 1874 and 1882, the results were 
obtained independently by the French, by the German, by the English, and by the 
American observers, each from a comparison of their own national expeditions. 
Really international, however, was the co-operative observation of the minor planets 
Iris, Victoria, and Sappho, organized by Gill (1888-9) ; and so was the well-planned 
and fruitful Eros campaign of 1930-1, in which forty observatories co-operated. 
Of this same "first type" is also the International Latitude Service, created in 1898 
and now centralized at Turin; it has become clear that this will have to be extended, 
since the number of co-operating observatories is not sufficient for the acquiring of 
complete information. Curiously enough, recent eclipse expeditions for the deter- 
mination of longitudes by the observation of the beginning totality have nearly 
always been made by several parties of one and the same nation; it seems that 
international co-operation for this subject has not been considered as yet. Recent 
co-operative schemes are concerned with the continuous observation of solar pheno- 
mena: the sudden disturbances, especially the flares, are studied by a number of 



M. MlNNAEKT 7 

observatories, located at different geographical longitudes, each observatory being 
responsible for a daily one-hour watch. The progress of radiophysics has necessitated 
a similar organization for the observation of the radio noise-bursts and outbursts, 
though in this case a smaller number of stations is sufficient, because these phenomena 
are automatically recorded, independently of the weather. Three or four transmitting 
stations are broadcasting daily in code form a survey of the solar activity, the iono- 
spheric perturbations, and the cosmic radiation in the last 24 hours ("ursigrams"). 

A second type of international co-operation has been organized in those cases 
where a scientific enterprise involves such a great amount of work that it could not be 
handled by one observer alone. Such collaboration is especially important for 
astronomy, which deals with an immense amount of material while depending on a 
limited number of scientists and instruments. Probably the earliest attempt in this 
direction was the conference convened by von Zach at Lilienthal, in the autumn of 
1800, where twenty-four astronomers of different nations assumed the task of under- 
taking a systematic search for the hypothetical planet between Mars and Jupiter. To 
the same category belong : the project of the Catalogue of the Astronomische GesehV 
chaft, initiated by Argelander in 1867 ; the celebrated enterprise of the Carte du Ciel 
(Paris, 1887) ; and, more recently, the Plan of Selected Areas (1905). In the theoreti- 
cal field, computational work is sometimes so extensive that even there a co-operation 
between the computing centres of different countries has proved very effective. For 
several tables of the astronomical almanacs the computations are carried out either 
in England, in Germany, in Prance, in Spain, in the U.S.A., or in the U.S.S.R. and 
the results are afterwards exchanged. Quite recently, the International Astronomical 
Union has planned a very interesting co-operative calculation of fundamental data 
concerning stellar atmospheres : the ionization, the absorption coefficient, and similar 
data will be computed at increasing depths for a great variety of model atmospheres, 
differing in their chemical composition; a dozen observatories have taken an interest 
in this work. The assistance of international computing centres with electronic 
machines will be of great value for similar problems in the future. 

Mutual co-operation in industry leads to specialization and to distribution of the 
work. The same has proved true in astronomical research. The modern big telescopes 
produce photographic records at such a tremendous rate that it would be impossible 
for the astronomers of such an observatory to run their instruments continuously and 
to work out all these invaluable documents. It has become more and more frequent 
that colleagues of all nations receive the photographs which are necessary for their 
work or get the opportunity to take them themselves, while the measurements, the 
microphotometrical investigation of the plates and the theoretical development are 
made at their own institutes. Such a co-operation was inaugurated by Kapteyn 
and Gill, when they agreed that the plates of the southern skies, obtained at the 
Cape Observatory (1885-90), would be measured at Groningen. The result was the 
Cape Photographic Durchmusterung. As one of the very numerous recent examples 
we may perhaps quote the Photometric Atlas of the Solar Spectrum, a result of a 
co-operation between the Mount Wilson and the Utrecht Observatories, and from 
which a Catalogue of Fraunhofer lines is being derived by another co-operation 
between some American and Dutch astronomers specialized in a particular branch 
of this subject. 

A third type of co-operation has a more centralized character. The co-operation of 
a great number of observations is entrusted to a central bureau, which makes the 



8 International co-operation in astronomy 

combined results available to all. In about 1850 Wolf, at Zurich, had already begun 
his computation of sunspot relative numbers from observations made in many 
countries. We have now a centre for the observation of solar flares at Meudon ; a centre 
for solar radio noise data at Sydney-Canberra; the Zurich centre for the study of 
sunspot development and for the publication of a Quarterly Bulletin of Solar activity ; 
the Cincinnati, Heidelberg, and Leningrad centres for the ephemerides of minor 
planets; the Moscow Catalogue of Variable Stars. The same principle is applied to 
astronomical bibliography, embodied in the Astronomischer Jahresbericht at 
Heidelberg, the Bulletin Analytique, and the Science Abstracts, and also to the 
publication of the volume Observatoires et Astronomes, by the Uccle Observatory. 
An important institution of this kind is also the central Bureau for Astronomical 
Telegrams, which has been working at Copenhagen since 1922, and communicates by 
wire and by circulars the latest discoveries about novae, comets, or exceptional 
planetoids. The Bureau International de l'Heure, set up at Paris in 1913 and re- 
organized in 1919, intercompares the time determinations of several observatories 
and broadcasts time signals of high precision. 

A far reaching plan for co-operative enterprise is the project of an international 
observatory, to be erected in the south of Europe, in excellent climatic conditions, and 
to be financed by those nations which desire to have the benefit of the telescopes 
there erected. This proposition was originally presented by the Polish delegation at 
the preparatory Copenhagen conference in 1947, and afterwards more fully developed 
by Professor Shapley, preference being given to a location in the southern hemi- 
sphere. It was put on a list of similar projects, submitted to the UNESCO, 
but relatively to these it was not considered to merit a high priority. In the mean- 
time, a plan of the same character, though less ambitious, had been realized in 
Switzerland, where it became a decided success. The high altitude station at 
Jungfraujoch was organized there, not only for astronomical research, but also for the 
study of biological and cosmic ray problems ; it receives contributions from several 
countries and is gradually developing its instrumental equipment in a most promising 
way. Very recently, a number of European observatories which had planned the 
establishment of small southern stations, have considered the possibility of building 
in common effort a first-class observatory in the southern hemisphere. 

Compared to these ambitious enterprises, a fourth category of international co- 
operative work might look quite insignificant : the unification of notations, termi- 
nology, and units. However, all actively working astronomers know the confusions 
which arise if there is no general accord on such matters. Quite recently the defini- 
tions of time had to be again modified, a new unit of Fraunhofer line strength had to 
be introduced and a new terminology for radio-astronomy had to be found, these 
innovations being made necessary by new scientific advances. It was also important 
to reach agreement on the normals of wavelength to be used in spectroscopic work ; 
or on the frequencies at which the solar radio noise will be observed, so that the 
measurements become comparable. 

3. Organizations foe International Co-operation in Astronomy 

The increasing international co-operation in astronomical work made necessary 
the organization of congresses and the creation of a body where such co-operation 
could be systematically planned. One of the first astronomical meetings with an 
international character was the congress at Gotha in 1796. Because of the presence 



M. MlNNAEBT 9 

of the Frenchman Lalande, the Austrian astronomers were not allowed by their 
government to take part at this meeting, and the court at Gotha was warned: 
"il pourrait bien s'agir d'autres revolutions que des revolutions celestes". The Duke and 
Duchess of Gotha did not care very much and were personally present. In 1863 the 
German astronomers founded the Astronomische Gesellschaft at Heidelberg, which 
from the start had a more or less international character, due to the membership of 
numerous foreigners. In 1904, systematic co-operation was started in the new field 
of astrophysics, when Hale succeeded in founding an International Union for 
Co-operation in Solar Research. The first meeting was held in St. Louis, in connec- 
tion with the International Congress of Science. In a remarkable speech he empha- 
sized that in co-operative scientific work special importance should be attached to 
the encouragement of individual initiative, no less than to the accomplishment of big 
projects for routine work. 

The Union for Solar Research gave the inspiration for the constitution of a much 
more far-reaching and systematic organization of astronomical research. Immedi- 
ately after the First World War, three meetings were held at London, Paris, and 
Brussels (1918-19), where an International Research Council was created, with 
several International Unions for the various sciences. Originally, the foundation 
of the Unions was not laid in a truly international spirit : interallied and neutral 
countries only were allowed to adhere. This may be frankly recognized: the Union 
has amply corrected this vitium originis, and "... 'tis thirty years since !" 
Already at the first meeting of the International Astronomical Union at Rome, in 
1922, Professor Certjlli opened the general assembly by a speech in which he em- 
phasized the necessity for uniting the astronomers of all nations, without any 
exception. After some years of hesitation, limitations to the membership were 
removed in 1926 and invitations for co-operation were addressed to Germany, 
Austria, and Hungary. But the seed of resentment bears evil fruits. It was only in 
1947 that Hungary became a member, and it was 1952 when Germany and Austria 
joined. There could be no clearer demonstration that scientific co-operation must be 
kept outside all political implications, that it should never be used as a means of 
uniting one group of nations against another group. The I.A.U. has quickly developed 
a considerable and very stimulating activity. There are now thirty-three member 
countries. The work is distributed over forty-two Commissions, which act in a most 
efficient way and may truly be said to give inspiration to individual initiative, 
according to Hale's recommendation. The successive meetings of the Union, separ- 
ated by intervals of three years, are each the result of intense scientific work and at 
the same time the starting point of new research. Besides these general meetings, 
small symposia for specialists on selected subjects have proved of great use. 

In this whole organization, the General Secretary is the man carrying the heavy 
burden and the greatest responsibility ; we shall never forget how Professor Stratton 
devoted more than ten years of his life to this important task and contributed more 
than anyone else to the vigorous development of the Union. 

The international feeling has now become so strong among astronomers that very 
quickly after the Second World War scientific co-operation was resumed, and no 
new dissensions were allowed to disturb this work. It is a source of pride and happi- 
ness to the members of the Astronomical Union that among them the Russian, Polish, 
and Czech astronomers as well as the American colleagues give each other a full- 
hearted co-operation, that they regularly exchange their publications and fraternize 



10 International co-operation in astronomy 

at their meetings. A critical event was the preparation of the 1952 meeting, for which 
a Russian invitation had been received but finally was not accepted because of the 
international tension. Neutrality was saved, for the decision taken applied in the 
same way to meetings in the U.S.A. and in the U.S.S.R. But it is a sad thought that 
a more positive demonstration of the universality of science could not yet be 
realized. 

International co-operation is more than the planning of a common scientific 
programme. It must never be forgotten that scientific research is made by men, and 
therefore it is not sufficient to exchange ideas — the scientists themselves should 
meet and discuss and work together. This is the reason why visits of astronomers to 
their colleagues abroad are so highly important and have developed to such an 
extent in recent years. It is a privilege when we are able to welcome great astro- 
nomers from distant parts of the world, to have them working, lecturing, discussing 
among us. But it is perhaps equally wonderful that young astronomers, in the spring- 
time of their life, are able to visit foreign observatories, to enjoy the stimulus of fresh 
contact and new surroundings, to see excellent astronomers at work in their institutes 
and in the midst of their collaborators. Temporary assistantships for foreigners 
have become available at many observatories. If the financial means of an institute 
are limited, an exchange may be easily organized between a junior staff member and 
one of his colleagues at another institute, the salary of each of them being available 
for the maintenance of the visiting astronomer. The recent success of such arrange- 
ments is in a considerable degree due to the activity, practical spirit, and never-failing 
helpfulness of Professor Stratton, now the president of the I.A.U. Commission for 
Exchange of Astronomers. 

In all international co-operation, the language differences are a major difficulty. 
At one time, Latin was the only vehicle of science and the common tongue of 
scientists all over the world. However, in the eighteenth and the beginning of the 
nineteenth century, when the influence of science on economic life and on society at 
large became increasingly important, the cryptic language of the learned was felt as 
an unendurable barrier between them and the nation. Temporarily, French was 
used in the learned societies all over Europe; then the vernacular superseded the 
Latin. But how about international relations ? At first, French, German, and English 
were the languages of the great scientific periodicals and standard works. Gradually 
science developed in many other countries, which began to publish in Italian, 
Spanish, Japanese. Recently the remarkable rejuvenation of astronomy in Russia 
has put the problem before us in an acute form. While science is growing, it requires 
■more and more labour to master the established disciplines and less time is left to 
learn foreign languages. There is thus an increasing need for reconsidering the whole 
problem, and radical solutions, such as the introduction of Esperanto as an auxiliary 
scientific language, should be seriously examined. Let the sceptical reader ask 
himself, whether he is able to propose a better solution. 

In all its varied ways international co-operation in astronomy has developed quite 
naturally out of the requirements of scientific life itself. It has adapted itself to our 
modern way of living, it has become increasingly important, even to such an extent 
that it really could not be done without anymore. Let us for a moment ponder about 
the significance of this international contact as a complement to national differentia- 
tion. It is sound that science should develop within each country as a part of the 
national activity and in narrow connection with the local circumstances, the produc- 



P. Th. Oosterhoff 11 

tion, the special interests of the people. By the selection of the problems, by the 
philosophical background on which these are treated, by the special qualities of the 
nation, it will have a character of its own, even if it is an "exact" science like astro- 
nomy. The interaction of the work of all these nations automatically eliminates errors 
due to preconceived ideas ; refinement of mind is complemented by stubborn labour 
or by practical common sense. But it is the faith of our life that ultimately there will 
never be contradictions between the findings of all these scientists, varied in their 
personalities and nationalities, since Nature is unique and since Truth is unique. 
And, finally, there emerges, in a purified form, what we may call International 
Science, which is no more the work of individuals but the work of the community, 
of humanity as a whole, the noblest expression of the human mind. 

International scientific co-operation demonstrates to all that there is a way of 
living together on Earth in peace and mutual aid and happiness ; a way of living 
which has not been found by politicians, but which has developed out of the simple 
desire for truth, and because we relied upon each other, and because we loved 
each other. 

. . . Ye Heavens, whose pure dark regions have no sign 
Of languor, though so calm, and though so great 
Are yet untroubled and unpassionate : 
Who though so noble share in the world's toil, 
And though so task'd keep free from dust and soil . . . 

Matthew Arnold, 1852. 1 



The International Astronomical Union 

P. Th. Oosterhoff 

Sterrewacht Leiden, Netherlands 



International co-operation in astronomy as well as international organization of 
certain specific problems which are too extensive or costly to be undertaken by a 
single observatory, have a long history. In the previous article Professor Minnaert 
has given a general outline of the historical development of international co-operation 
in astronomical science. The same topic was treated by Professor Stratton when he 
delivered his Presidential Address at a meeting of the Royal Astronomical Society. 2 
Nevertheless it was not until 1919 that the many different and separate efforts 
were combined into a single organization. In common with some other Unions the 
International Astronomical Union was founded at the Constitutive Assembly of the 
International Research Council, which was held at Brussels in July 1919. During the 
thirty-four years which have elapsed since its establishment the Union has developed 
a great many activities in different fields and it has no doubt become an institution 
of unique value for astronomical science. Interesting facts about the early history 
of the I.A.U. can be found in an article by Dr. W. S. Adams (Publ. Astron. Soc. 
Pacific, 61, 5, 1949). 

„' For ,^11 text see the fascinating collection "Dichters over Sterren" by M. Minnaert (van Loghum Slaterus, Arnhem, 
219 pp., 1949). — The Editor. ' ' 

2 F. J. M. Stkatton; AJ.1V. 94, 361-372, 1934. 



12 



The International Astronomical Union 



This development however has not always been smooth and in the course of years 
many difficult problems had to be solved. Most of these problems paralleled the 
international political situation. It is very encouraging that the Union despite these 
political setbacks, is to-day as strong as, and probably even stronger than it ever was 
before. This result is not only due to the efforts, the good- will and the wise decisions 
of the members of the Executive Committee, who had a large share in determining 
the general policy of the Union, but also to the conviction of all the individual mem- 
bers that a strong international organization like the Union is essential for a sound 
and fruitful development of science in general and of astronomy in particular. Many 
astronomers of widely different nationalities have spoken to this effect and have given 
expression to their belief that the international relations between astronomers are 
too strong to be broken or even affected for long by political forces. We may quote 
here from a message by Sir Arthur Eddington which was read at the concluding 
meeting of the General Assembly of Stockholm in 1938: 

"But, if in international politics the sky seems heavy with clouds, such a 
meeting as this at Stockholm is as when the sun comes forth from behind the 
clouds. Here we have formed and renewed bands of friendship which will 
resist the forces of disruption." 

This firm belief has never changed and it has succeeded in making the Union the 
strong organization it is now. 

Before dealing with the present-day activities and problems of the Union, we shall 
pay tribute to those astronomers who have given much of their thought and time for 
the benefit of the Union, and we shall give some numerical data to demonstrate the 
continuous growth of the Union. 

The Union is governed by the Executive Committee, consisting of a President, 
five Vice-Presidents, and a General Secretary who is also the Treasurer. The Presi- 
dent stays in office for one term, normally of three years, whereas the other officers 
are elected for two terms. The Executive Committee is responsible for the adminis- 
tration of the affairs of the Union and acts in accordance with the decisions of the 
General Assembly. As a rule the General Assembly meets once every three years. 
For obvious reasons it has not always been possible to keep to this rule and altogether 
eight General Assemblies have been held so far. Without derogating the great merits 
of the many Vice-Presidents who have served the Union, we give here a list of the 
Presidents and General Secretaries who have been in office since the foundation of 
the Union. 



Presidents 
Mr. B. Baillaud 
Prof. W. W. Campbell 
Prof. W. db Sitter . 
Sir Frank Dyson 

Prof. F. SCHLESINGER. 

Prof. E. Esclangon . 
Sir Arthur Eddington 
Sir Harold Spencer Jones 
Prof. B. Lindblad 
Prof. O. Struve 



General Secretaries 



1919-1922 
1922-1925 
1925-1928 
1928-1932 
1932-1935 
1935-1938 
1938-1944 
1944-1948 
1948-1952 
1952- 



Prof. 


A. 


Fowler 


1919-1925 


Prof. 


F. 


J. M. Stratton 


1925-1935 


Prof. 


J. 


H. Oort 


1935-1948 


Prof. 


B. 


Stromgren . 


1948-1952 


Prof. 


P. 


Th. Oosterhoff 


1952- 



Among the names on these lists two stand out clearly because of their very long 
service. They are Prof. F. J. M. Stratton, who fulfilled the task of General Secretary 



P. Th. Oostebhofp 13 

for ten years, over three consecutive terms, and Prof. J. H. Oort, who kept the same 
office for the usual two terms, but on account of the war these terms covered a period 
of thirteen years. It can safely be said that these two prominent astronomers, who 
have sacrificed on behalf of the Union so much time which they might have used for 
their own research, know the Union and its affairs better than anyone else. Their 
prudence and careful handling of the Union's affairs have contributed greatly to the 
continual growth and strength of this organization. For this work alone they have 
earned lasting gratitude from astronomers all over the world. We shall see later that 
Professor Stratton did not discontinue in 1935 his efforts to further international 
collaboration and international scientific organization. 

The growth of the Union can be demonstrated by some simple figures. 





Number of 




Number of 




standing 


Number of 


adhering 


Year 


commissions 


members 


countries 


1922 


32 


207 


19 


1938 


31 


553 


26 


1952 


39 


814 


33 



The figures indicate the situation just after the first General Assembly at Rome, 
after the sixth at Stockholm and after the eighth and latest at Rome. The figures in 
the last column are especially important, showing that the adjective "international" 
is continually gaining weight. The figures in the third column prove that astronomy 
is very much alive at present — though they may cause some worry to the General 
Secretary. The table, however, gives a very incomplete picture of the growth of the 
Union. Although the increase in the number of commissions already indicates an 
extension of its activities, a number of important developments have taken place 
especially since the end of the war. 

Although some of the regular activities of the Union, many of which were started 
immediately after its foundation, have been enumerated in the preceding article by 
Professor Minnaert, they are too important to omit for this reason in an article 
which should give a description of the I.A.U. Much work has been done by the 
standing commissions and it is difficult to estimate how many times discussions 
between members of these commissions have paved the way for international 
collaboration. In this connection should be mentioned the stimulating influence of 
the Union on work bearing upon the "Carte du Ciel" and the plan of Selected Areas. 
The work on nomenclature, the co-ordination of solar research, with publication at 
regular intervals of solar observations, the work on ephemerides and Minor Planets, 
the naming and cataloguing of variable stars are only a few of the items which require 
international collaboration and which therefore have received the full attention, 
sometimes including financial aid, of the Union. Furthermore the Union is scienti- 
fically and financially concerned in a number of permanent services, like the Inter- 
national Latitude Service, the International Time Bureau, and the telegram Bureau 
at Copenhagen. The first two of these services work under the auspices of the I.A.U. 
and of the International Union of Geodesy and Geophysics. The Union also took a 
very active interest in the International Longitude Campaigns of 1926 and 1933, and 
it is now engaged with other Unions, under the auspices of the International Council 
of Scientific Unions, in the preparations for an International Geophysical Year and 
another Longitude Campaign. 



1* The International Astronomical Union 

Of the developments which became important after the end of the last war we 
should mention first the organization of an increasing number of international 
symposia, some of them taking place at the occasion of a General Assembly, others in 
the interval between such meetings. It is not unusual for a considerable part of the 
meetings of the standing commissions during a General Assembly to be devoted to 
rather technical matters and the discussions are naturally restricted within the 
province of the commission. Consequently the introduction of symposia opened new 
possibilities for an exchange of opinions on more general scientific topics. Since the 
war the following symposia have been held : 

During General Assembly, 
Zurich, 1948 

Infra-red Spectrophotometry 

The Spectral Sequence and its Anomalies 

The Abundances of the Chemical Elements in the Universe 

Paris, 1949 

Problems of Cosmical Aerodynamics 

(Organized by the International Union of Theoretical and 

Applied Mechanics and the I.A.U.) 

During General Assembly, 
Rome, 1952 

Stellar Evolution 
Astrometry of Faint Stars 
Astronomical Instrumentation 
Spectra of Variable Stars 

Groningen, 1953 

Co-ordination of Galactic Research 

Cambridge {England), 1953 

Gas Dynamics of Interstellar Clouds 
(Organized by I.A.U. and I.U.T.A.M.) 

These symposia have proved so successful that it may be assumed that the Union 
will continue to organize them. 

The other most important development, though of a completely different character, 
was the establishment of UNESCO, which has taken an active interest in inter- 
national organization. Since 1947 the Union has received annually considerable 
sums from this body, which have enabled the Union to increase its activities to a 
great extent. The grants allotted by UNESCO to the I.A.U. are as follows : 



1947 


11,740 


1948 


21,880 


1949 


14,000 


1950 


13,105 


1951 


17,900 


1952 


14,300 


1953 


13,750 



P. Th. Oosterhoff 15 

As the income from the annual dues paid to the Union by the adhering countries 
amounted to $24,018 in 1951 and to $23,451 in 1952, it is clear that the financial aid 
from UNESCO plays an essential role in the budget of the Union. The International 
Latitude Service, le Bureau International de l'Heure, several symposia, many 
publications and other activities of the Union have profited from this financial 
aid. 

This larger budget led also to the formation of Commission 38 for the exchange 
of astronomers. Under the able presidency of Professor Stratton this commission 
has been very active and many young and promising astronomers have obtained 
through the efforts of Professor Stratton and his colleagues the opportunity to work 
for shorter or longer periods at renowned institutes in foreign countries. This form 
of exchange is so important and has proved so successful that the Union will con- 
tinue to support Commission 38 financially, even though UNESCO has discontinued 
its financial aid towards this special purpose. 

It is realized that this survey of the activities of the Union is very incomplete. 
The choice of the subjects treated is a personal and rather arbitrary one, but a fuller 
treatment would go beyond the scope of this article. However, some of the prob- 
lems which the Union has still to solve should be mentioned. The first of these 
difficulties is probably met in all international organizations. It is the problem of 
languages. The fact that many tongues are spoken by the members of the Union 
usually does not seriously hamper conversation and the verbal exchange of ideas 
between them, as often a language can be found which is understood by both parties. 
But in large meetings, like General Assemblies, meetings of commissions and sym- 
posia, long translations are sometimes required which impede an effective and 
expeditious course of the proceedings. The problem is most urgent however with 
respect to scientific publications. Here the Union has always stressed the desirability 
that abstracts in another main language be added to the original articles. It seems 
impossible that the Union can solve this problem of languages completely. It will be 
the task of several generations to come to remove the barriers which at the present 
time impede the free intercourse between the different nationalities. 

The second problem which may be mentioned here is one of an "organizational" 
character. Since its establishment the Union has divided its scientific task over a 
number of standing commissions. It is evident that any subdivision of astronomical 
science into a number of specialized fields must be arbitrary and artificial. Probably 
for this reason some other big Unions have followed a less drastic course and work 
through a small number of large sections. During its early history the number of 
memt/ers of the Union was so small that the standing commissions could be con- 
sidered as working groups of a relatively small number of experts in the field. At 
present an astronomer can only be a member of the Union if he is nominated as 
member of a commission. As a consequence of this rule and of the present growth of 
astronomy, several commissions have become very large, and it is well known that 
the efficiency of a commission is certainly not proportional to the number of its 
members. The difficulties indicated here have been studied and discussed more than 
once by the Executive Committee, but no other form of organization has been sug- 
gested so far which would guarantee an improvement over the present system. 
During the last General Assembly a number of combined commission meetings were 
arranged which proved to be quite useful and Professor Lindblad suggested that in 
large commissions small working parties should be formed. Although we do not know 



16 The International Astronomical News Service 

what the future course of the Union will be, it is reasonable to expect that the Union 
which has already made itself indispensable to astronomy during the few decades of 
its existence, will succeed in solving such organizational difficulties. 

This article would be incomplete without a few words about the International 
Council of Scientific Unions (I.C.S.U.). This body, the central organization of the 
International Scientific Unions, was established in 1931 as a continuation of the 
International Research Council, which was founded by the Allies in 1919 after the 
First World War. At present, eleven Unions adhere to this Council. Space does not 
permit describing in any detail the structure and the activities of this important 
organization. Although I.C.S.U. and UNESCO are both completely autonomous 
organizations, an agreement of mutual recognition was drawn up between them in 
1947. As a consequence, the granting of financial support by UNESCO to the 
International Scientific Unions adhering to I.C.S.U. takes place through the inter- 
mediary of I.C.S.U. Furthermore, I.C.S.U. has organized and finances a number of 
special research stations of which the High Altitude Research Station at the Jung- 
fraujoch has proved to be of great importance for astronomical research. Another 
form of activity of I.C.S.U. is the formation of Joint Commissions between members 
of two or more Unions in order to co-ordinate efforts in special fields which fall 
within the domain of more than one Union. At the moment, members of the I.A.U. 
are active in four of these Joint Commissions, viz. on High Altitude Research 
Stations, the Ionosphere, Solar and Terrestrial Relationships, and Spectroscopy. 

The board of I.C.S.U. consists of a Bureau, an Executive Board, and a General 
Assembly. No doubt it will interest the reader to know that Professor Stratton, 
who has served the I.A.U. for so many years, has also been the General Secretary of 
I.C.S.U. from 1937 until 1952. Few scientists have taken such a very large share in 
the efforts to improve international organization and collaboration as has Professor 
Stratton. 



The International Astronomical News Service 
Julie Vinter Hansen 

Universitetets Astronomiske Observatorium, Copenhagen 
Summary 

This is the story of the development of the International Astronomical News Service from the wishful 
thinking of Tycho Brahe up to the present organized global news service by telegrams and circulars, 
under the auspices of the International Astronomical Union. The location of the present International 
Telegram Bureau is : University Observatory, Copenhagen, Denmark. 



The feeling of the importance of co-operation between scientists has been steadily 
growing through the years. No doubt most scientists have always felt an urge to 
contact other learned men to discuss problems, discoveries, and inventions. In 
ancient times such contacts were not easily made; travelling was hazardous and 
mail-service non-existent, hence progress was slow. The lack of intercourse between 
scientists sometimes led to bitter quarrels about the priority of theories, discoveries, 
or inventions, and it was not easy to pass judgment on the various claims. 



Julie Vinteb, Hansen 17 

In astronomy we see how the discoveries and calculations of the ancient Eastern 
countries spread to Europe where particularly the early Greeks developed theories 
about the motions of the celestial objects around us, and this accumulated knowledge 
was kept alive by the Arabs during the decline of science in Europe during the 
Middle Ages. With the Renaissance astronomy, as all other sciences and arts, awoke 
to new life and eminent astronomers even voiced ideas of organized co-operation. 
Tycho Brahe, for instance, in his description of his life* and work (written about 
1598) does some wishful thinking in this respect. Writing about his own catalogue of 
1000 stars, he points out the desirability of observing the Southern stars — both those 
visible from Egypt and those further South — and he adds: "So if some mighty 
nobleman would care to fulfil our own and others' wishes in both these respects, they 
do a very good deed that would be ever gloriously remembered. Up to now no one 
has even tried to do a thing like this in the right way, let alone carried it out, as far 
as I know. I would be willing to provide the necessary instruments and tools if 
somebody could organize the work and get the right people for such a deserving 
enterprise". A little later in the same paper, talking about the importance of deter- 
mining correct geographical positions of localities on the Earth, he once more calls 
upon "kings and princes and other mighty noblemen in widely separated parts of the 
world" to make suitable and generous preparations in this respect for "then they 
would really be doing a good deed, and in this way astronomy, which is in need of 
widely different terrestrial horizons, would develop towards greater perfection". 
It took over 200 years before any organized work as that proposed by Tycho 
Brahe was undertaken. That Tycho also keenly felt the importance of closer inter- 
course between fellow astronomers is seen from a paragraph in the same paper in 
which he tells that before King Frederik II of Denmark offered him the island 
Hven as site for an observatory he had had plans for settling in Basel, one of the 
reasons for this choice being, that "Basel is located so to speak at the point where the 
three biggest countries in Europe, Italy, France, and Germany meet, so that it would 
be possible by correspondence to form friendships with distinguished and learned 
men in different places. In this way it would be possible to make my inventions more 
widely known so that they might become more generally useful". Into these remarks 
may be read a desire for a centre, from which useful astronomical news could be 
distributed. Such a centre, however, took its time to materialize. 

The first start was made in Denmark in connection with a comet medal that the 
Danish King Frederik VI instituted on 17th December, 1831,f this gold medal to 
be given to the person who was the first to announce the discovery of a new telescopic 
comet. All discoverers had to write immediately about the new comet to H. C. 
Schumacher, professor of astronomy in the University of Copenhagen, but resident 
in Altona, Holstein, and editor of the Astronomische Nachrichten. In case of more 
than one claimant Schumacher was to decide who was to be given the medal. The 
first of these medals was presented in January, 1833, to Gambart, Marseille, for the 
discovery of a comet on 19th July, 1832 (Gambart, 1833). 

A few years afterwards co-operation was established with Francis Baily (1835), 
in such a way that discoverers of comets "if in any part of Europe except Great 
Britain must send immediate notice to Professor Schumacher of Altona ; and if in 



, ^?„ Tycho Brahe ' s description of his instruments and scientific work, Bet. Kgl. Danshe Videnskabernes Selskab, Copenhagen, 
1946. 
t Announcement published in Astron. Nachr., 10, No. 221, 1832. 

3 



18 The International Astronomical News Service 

Great Britain, or any other part of the globe except Europe, must send immediate 
notice to Francis Baily, Esq., of Tavistock Place, London. Professor Schumacher 
and Mr. F. Baily are to determine whether a discovery is to be considered as estab- 
lished or not ; but should they differ in opinion, Dr. Olbers, of Bremen, is to decide 
between them". This arrangement seems to have worked very well. 

In addition to publishing news of astronomical discoveries in the Astronomische 
Nachrichten, Schumacher also printed, when needed, circulars to spread the news of 
such discoveries as quickly as possible. These circulars contained not only dis- 
coveries of comets and cometary orbits and ephemerides but also news about asteroids, 
for those were the days when the discovery of a minor planet was considered quite a 
remarkable event. Famous names like that of Gauss are sometimes found among the 
contributors to the circulars, and these circulars were not always as matter of fact as 
present-day circulars from the Central Bureau for Astronomical Telegrams. For 
instance, in a circular of 22nd October, 1847 (reprinted in A.N., 26 No. 616), 
announcing the discovery by Hind of a minor planet, a proposal by Sir John F. W. 
Herschel to name the next discovered asteroid Flora is mentioned, and Schumacher 
quotes in full the following lyrical paragraph from Herschel's letter: 

"Pallas, Juno, Ceres, and Vesta, as sober and majestic Duennas will abundantly 
provide for the respectability of the group between Mars and Jupiter, while 
Astraea, Iris, Hebe, and Flora will attract all eyes and fill all imagination with 
sweet and graceful images". 

A quicker way to spread astronomical news was provided when telegraph service 
came into being. In 1869 the Imperial Academy of Sciences in Vienna instituted a 
gold medal or the corresponding value thereof to be given to the person, who first 
announced a new telescopic comet. The announcement was to be sent, preferably 
by telegram to Vienna and from there the news was to be distributed by telegrams 
to selected observatories. The first two medals went to Tempel (1873), of Marseille, 
for comets discovered in 1869. In 1873 the intercourse between astronomers on 
opposite sides of the Atlantic Ocean was made more easy when the Associated 
Trans-Atlantic Cable Companies granted the Smithsonian Institution of Washington 
a limited number of astronomical cablegrams free of charge between America and 
Europe, more strictly between the Smithsonian Institution and the Astronomer 
Royal of England.* In the first place the idea was to transmit discoveries of comets 
and planets, but laterf it was stated that other astronomical phenomena, for instance, 
disturbances on the Sun, outbursts of some variable stars, unexpected meteor 
showers, would be proper subjects for telegrams. Although great satisfaction was felt 
in this quick new way of obtaining news from distant observatories, no real organized 
news service was yet in existence. 

At a congress of the Astronomische Gesellschaft in Berlin, Professor Forster 
(1879) pointed out that the existing arrangement for the distribution of astronomical 
news was not quite satisfactory, particularly the form of the telegrams did not permit 
any check on the correctness of the imparted numbers except through repetition; 
also national vanities had showed up. Forster thought it desirable that a central 
news bureau be established. Although astronomers took kindly to this idea, nothing 
much was done about it. In 1881 Forster once more aired his ideas about such a 



* Announcement in M.N., 33, 369, 1873. 
t Announcement in M.N., 34, 185, 1874. 



Julie Vinter Hansen 19 

bureau. He considered the telegraphic transmission of astronomical news free of 
charge a doubtful blessing, because experience showed that people when not having 
to pay were apt to be too hasty in transmitting their supposed discoveries without 
checking them carefully. He also wished for a telegraphic code that would permit 
immediate check on the cabled figures. At the congress of the Astronomische 
Gesellschaft in Strasbourg, in 1881, Forster's plans for a central bureau were 
debated (Forster, 1881). Some favourable resolutions were passed, and there the 
matter rested — presumably because some astronomers were reluctant, considering 
the proposals too radical and fearing national vanities might be hurt. 

In 1882 the appearance of the bright September comet brought on a crisis and 
made the astronomers realize that the arrangement for the transmission of news was 
thoroughly inadequate. Forster (1882) took the initiative and had a central bureau 
established in the German town of Kiel, under the management of Professor A. 
Krueger, the editor of the Astronomische Nachrichten. This bureau had at its start 
thirty-nine subscribers, thirty-eight in Europe, one in Asia (Tashkent). For the 
U.S.A. Ritchie, Boston, was to act as intermediary between American observatories 
and the bureau. The first telegram code used was the so-called "Science Observer 
Code", a code that had been warmly recommended to the Astronomische Gesellschaft 
at the meeting in Strasbourg by S. C. Chandler and John Ritchie, the editors of 
Science Observer. This code was in use in the U.S.A. and it is based on a dictionary: 
A Comprehensive Dictionary of the English Language, by Joseph E. Worcester, 
L.L.D., Boston, 1876. In this code it is possible to communicate five-figure numbers, 
and thus telegraph positions or orbits and ephemerides together with a check- 
number, being the sum of the preceding numbers; for instance, if w = 354° 9' 
= 35409 you had to seek the 9th word on page 354 of the dictionary and came out 
with the word pyrrhic. This code, however, was not quite satisfactory; some of the 
pages did not contain the needed 100 words, and none of them had a word for the 
figure 00, and furthermore several words listed in the dictionary were the same, 
so that uncertainties arose when decoding. Krueger (1882) therefore very soon 
decided to change code and adopted a five-figure code that in principle was the same 
as that still in use, although minor changes have taken place. This new code was an 
amendment of a figure code, proposed by Karlinski (1866), of Cracow. At the start 
the Vienna Observatory had agreed to help the Central Bureau in providing orbits 
and ephemerides to be telegraphed from the bureau to its subscribers. In 1883 the 
Harvard College Observatory undertook the distribution in the United States of 
astronomical information received from the bureau; for many years it favoured 
another code than the bureau in its telegrams, a syllabic code (Gerrish system). 

The new telegram bureau functioned well, not only by telegrams but also, in less 
urgent cases, through circulars. From a notice in the Astronomische Nachrichten, 
dated 5th December, 1883, about meteor observations, it can be read that Krueger 
had been in for the same worries that befell later leaders of central bureaus, namely, 
how much to convey by telegrams and to how many of the subscribers. All went 
smoothly until the fateful year of 1914, when World War I started, and telegraphic 
and postal communications between warring nations came to a stop. At that time 
Professor H. Kobold was head of the Central Bureau in Kiel and on 3rd November, 
1914, he made an arrangement with Professor Elis Stromgren, in neutral Denmark, 
to the effect that the Copenhagen Observatory took over the management of the 
Central Bureau for astronomical telegrams during hostilities. Elis Stromgren was 



20 The International Astronomical News Service 

well suited for this work, as in earlier years he had been an assistant in Kiel and was 
familiar with the workings of the Central Bureau. It was possible for him all through 
the war to keep up astronomical intercourse and satisfactory news service between 
astronomers all over the world. After the war when great bitterness existed between 
various peoples, it proved impossible to return to the status of having only one 
Central Bureau. The international relations were so strained that some astronomers 
even refused to deal with the Copenhagen Bureau, because it was considered too 
closely connected with Kiel. For a short while Professor B. Baillatjd, of the Paris 
Observatory, undertook to act as intermediary between such astronomers and the 
Copenhagen Observatory. In 1 9 1 9 the International Research Council was founded in 
Brussels and from this Council sprang the International Astronomical Union (I.A.U.), 
at first allowing only astronomers from the Entente powers to adhere, but soon 
astronomers from neutral countries were also invited to join. 

The I.A.U. established its own central bureau for astronomical telegrams in 
Uccle, with Professor G. Lecointe as its director. This bureau started work on 1st 
January, 1920, and three central bureaus were thus in existence: Uccle and Kiel, 
with Copenhagen acting as an intermediary between the two. This peculiar state of 
affairs lasted until 1st October, 1922, when the I.A.U. Bureau was transferred from 
Uccle to Copenhagen, with E. Stromgren as its director. The number of Central 
Bureaus were thus reduced to two, and friendly relations existed between these 
two bureaus. 

The Second World War brought the next great upheaval in the story of the 
astronomical news service. Denmark tried to stay out of the debacle but was invaded 
by the Germans in April, 1940. Elis Stromgren, being a great diplomatist and being 
well-known in German circles, was despite all odds fairly successful in keeping up 
international astronomical intercourse; he even obtained permit to send code 
telegrams, via the Lund Observatory, Sweden, to subscribers in allied countries, and 
the I.A.U. circulars, although often late in arriving, generally did reach their wide- 
spread destinations. The I.A.U. Bureau having functioned all during the war could 
at the end of hostilities immediately resume its activities in a normal way. The old 
Kiel Bureau did not fare so well; it had been transferred to the Astronomisches 
Rechen-Institut in Berlin, but in the present political difficulties of that city the 
location as news centre was most unfavourable, and the bureau was moved to 
Heidelberg with Professor Koppp as its director. When telegraph service was again 
established between Denmark and West Germany the two bureaus could once more 
co-operate in the old routine. 

Elis Stromgren remained director of the I.A.U. bureau in Copenhagen until 
his death in 1947, when the writer took over. The bureau is steadily acquiring new 
subscribers; at present it has sixty subscribers to its telegrams and circulars and 
eighty-five that receive circulars only. Its services really cover the whole of our globe 
from Moscow to New Zealand and from Tokyo all around the world to Rio de Janeiro 
and Santiago, Chile. In U.S.A. the Harvard College Observatory receives the 
I.A.U. telegrams and distributes the news to the Americas and in the U.S.S.R. the 
Sternberga Observatory, Moscow, likewise acts as intermediary for telegrams between 
the bureau and Soviet institutions. The I.A.U. circulars, however, go to numerous 
subscribers in both the above-mentioned countries. Since 1922 and up to the end of 
1951 the I.A.U. Bureau has transmitted 6409 single telegrams and 1339 circulars 
were printed. The bureau is supported by grants from the I.A.U. and from the 



Julie Vinter Hansen 21 

Danish Rask-Orsted foundation which greatly helps to reduce the costs of 
subscription.* 

Life at the bureau is quite variegated. At times when nothing much is happening 
in the sky, all is quiet ; at other times life grows very hectic when a bright comet or 
nova or several comets are discovered at practically the same time. The clerical work 
of decoding and rewriting the telegraphic messages is so well organized that in an hour 
or so we are able to have the telegrams to all our sixty subscribers ready to go to the 
telegraph office, but this clerical work represents the smallest part of the bureau 
work. The deliberations before it is decided to transmit a message often claim several 
hours. It has to be investigated whether the announced object really is new and 
whether the announcement may be considered trustworthy, and if the decision is 
favourable the question arises whether telegrams have to go to all subscribers or only 
to a selected few or whether circulars, perhaps sent by air-mail, will be sufficient. 
This probing of the news is by far the most important and difficult part of our work, 
and we certainly have our surprises, when for instance, an announced bright nova 
proves to be one of the major planets. Besides transmitting the news of discoveries 
in the sky the bureau also has the responsibility of following up this news, so that 
the new object may be properly investigated, that is, for comets we have to provide 
our subscribers with sufficient orbits and ephemerides. In busy times the staff of 
the Copenhagen Observatory is far too small to take care of this task but fortunately 
a number of astronomers and observatories come forward to help us. Very often the 
discoverer himself or his near colleagues take a patriotic pride in providing the 
necessary computations, and the Leuschner Observatory, particularly Dr. Cun- 
ningham, is also keen on such computations. For the short-period comets the 
Computing Section of the British Astronomical Association does extensive and 
appreciated work in supplying predicted elements and ephemerides. 

I may conclude in adding that the activities of the bureau are well-known outside 
the astronomical world, which in a way is gratifying, but also has its drawbacks; 
we can, for instance, thank non-astronomers for good observations of meteors but we 
have also had our share of announcements of "flying saucers" ; unfortunately it has 
not added to our popularity in a sensation-greedy world that so far we have staunchly 
refused to spread such announcements through the channels at our disposal. 



References 

Baily, F 1835 M.N., 3, 132. 

1870 Astron. Nachr., 76, No. 1809. 

Forster, W 1879 Vierteljahrsschrift d. Astron. Gesellschaft, 

14, 345. 

1881a Astron. Nachr., 100, No. 2386. 

1881b Vierteljahrsschrift d. Astron. Gesellschaft, 
16, 350. 

1882 Astron. Nachr., 103, No. 2472. 

Gambart, J. F. A 1833 Astron. Nachr., 10, No. 238. 

Karlinski, F 1866 Astron. Nachr., 66, No. 1562. 

Krueger, A 1882 Astron. Nachr., 104, No. 2481. 

Stratton, F. J. M 1934 M.N., 94, 361. 



* For a summary dealing with international astronomica co-operation, see also Stratton (1934). 



The Committee for the Distribution of Astronomical Literature 
and the Astronomical News Letters 

Bart J. Bok 

Harvard College Observatory, Cambridge, U.S.A. 

and 

V. KOURGANOFF 

Laboratoire d'Astronomie de Lille, France 



In 1940, shortly after the invasion of the Low Countries, it became evident that for 
several years to come there would be great obstacles to the continued exchange on a 
world-wide basis of astronomical literature. At the Wellesley meeting of the American 
Astronomical Society (September, 1940) the Council appointed a Committee for the 
Distribution of Astronomical Literature (C.D.A.L.), with H. R. Morgan and Jambs 
Stokley as members and Bart J. Bok as Chairman. The C.D.A.L. was charged 
with the responsibility of promoting as far as possible the continued world-wide 
flow of astronomical literature. Wilhblm Brunner, in Switzerland, Bertil 
Llndblad, in Sweden, Kathleen Williams, in Great Britain, and G. Neujmin, of 
the U.S.S.R., offered to assist in the work of this Committee. Much help was also 
received from J. H. Oort, in Holland, and A. Kopff, in Germany. About a dozen 
copies of each astronomical publication from the United States, Great Britain, and 
Canada were distributed in this fashion and in return the C.D.A.L. received from 
abroad many publications intended for distribution and circulation in the United 
States, Great Britain, and Canada. 

With the entry of the United States into the war (December, 1941), there developed 
increasing obstacles to the sending abroad of books and publications and in the 
summer of 1942 the C.D.A.L. decided upon the publication, in co-operation with the 
U.S. Department of State and the Office of War Information, of a monthly Astro- 
nomical News Letter, reporting on current research activities at home and abroad. 
These News Letters received a very wide distribution and provided for many 
astronomers abroad an invaluable continuing link with astronomical activity in parts 
of the world otherwise almost completely cut off. from communication. The News 
Letters proved also to be very helpful in keeping astronomers in Allied military 
service, or engaged upon military research, informed about the current happenings 
in their peace-time field of research. Altogether thirty-six of these war-time monthly 
News Letters were distributed before the project was abandoned. Some of our 
colleagues abroad, in Europe, South Africa, Australia, and the U.S.S.R., were 
extremely helpful in promoting the wide distribution of these News Letters. 

During the war years Brunner and Lindblad did everything possible to provide 
the C.D.A.L. with copies of current astronomical literature from the occupied 
countries and Germany. The few available copies of each publication were first 
abstracted, and the abstracts were then collected in a separate series of C.D.A.L. 
"Bulletins," which were then distributed in mimeographed form to interested 
astronomers in the United States, Canada, and occasionally Great Britain. 

22 



BART J. BoK AND V. KotTRGANOFF 23 

At the end of the war, the Council of the American Astronomical Society requested 
the C.D.A.L. to do what it could for the rehabilitation of devastated observatory 
libraries in war-torn countries. Through the services of an enlarged Committee of 
American astronomers and a group of representative astronomers abroad, the needs 
were assessed and a significant contribution toward the rehabilitation problem was 
made. Since the funds available to the C.D.A.L. were small, we had to depend 
largely on gifts from publishers and observatory directors, but help was freely given 
and most of the requests could be satisfied. 

When in 1948 all western-language abstracts of astronomical publications of the 
U.S.S.R. were discontinued, a new need developed and the Astronomical News Letters 
were resurrected, as the medium for the publication of abstracts in English of 
astronomical papers published only in Russian or in another language of the Soviet 
Union. Otto Struve became the Editor of the new series, which began with News 
Letters No. 37 and which has now reached No. 72; his principal collaborators in the 
preparation of the abstracts were A. N. Vyssotsky and S. Gaposchkin, with others 
occasionally assisting in the work. As in the past the mimeographing and distribution 
of the News Letters was handled by Margaret Olmsted at Harvard Observatory. 
Early in 1953, Struve had to resign as Editor and this post has now been taken by 
V. Kourganoff, who will report below in detail on current activities, and plans for 
the future. 

When I look back upon my fourteen years of intimate association with the 
C.D.A.L. and the News Letters, I am very much aware of the fact that the work 
could never have been done without the whole-hearted co-operation from the 
astronomical fraternity the world over. It is only because of this co-operation, 
freely given by all, that in astronomy we have had a record of which we can all be 
proud of continued exchange of information during and after the war. I have named 
already the astronomers and organizations who were principally involved in the 
work. But I should add here that success could never have been achieved without the 
enthusiastic support of Jean Rendall Arons, Elizabeth Chapman, and Margaret 
Olmsted, who successively acted as Executive Secretary to the C.D.A.L. 

It is interesting to reflect how low the actual financial outlay for the whole pro- 
gramme has been. During the past fourteen years, the costs of the programme have 
been met, first of all, through three grants from the Foreign Relations Division of 
the U.S. National Research Council; these total $700. In addition, we have received 
grants totalling $200 from the American Astronomical Society and quite recently a 
grant of $300 from the International Astronomical Union. Astronomers in the 
United States and Canada have paid a small subscription fee for the recent series of 
News Letters, but there have been no other sources of subscription income. All this 
goes to prove that the astronomical fraternity can do wonders with limited funds, 
provided the project is one in which all are ready to assist. 

B. J. B. 



The use of Latin in "learned works" at the time of Copernicus and Newton, when 
all cultured people could read this language, was very fortunate, and contributed in a 
most powerful way to the progress of science. 

No major difficulties seem to have been encountered, when French took progres- 
sively the place of Latin as the "official language" of scientists. 



24 The committee for the distribution of astronomical literature and the astronomical news letters 

Extension to German and English, in the nineteenth century, was smooth enough, 
and was facilitated by the increased interest in foreign languages for general purposes, 
including literature, trade, and travel. 

Thus a stable state had been reached at the beginning of our century when the 
knowledge of three "official" languages was sufficient for every scientist of any nation. 
Distinguished contributions from Dutch, Swedish, Danish, Norwegian, Italian, 
Russian, Japanese, Polish, Spanish, and other scientists, could get an international 
diffusion, being all published in French, German or English. 

Real difficulties begun in the problem of "linguistic barriers" when some nations 
started scientific publications in their own "unofficial" languages. This was, 
apparently, not only an effect of the development of the "national feeling", but also 
very often the result of a deep change in the structure of scientific research itself. 
For the progressive differentiation and the extreme specialization, made it increasingly 
difficult for those attracted to scientific research to maintain the old standard of 
general culture and to fit themselves to write good papers in any but their own 
language. 

So long as only isolated scientists broke with the tradition of "three languages", 
international co-operation was in no real danger. Authors who, for some reasons, 
did not obey the general rule knew very well that their paper would be "buried", and 
would receive less diffusion, but they took the risk. They often published, however, 
an abstract in French, German or English. 

The sudden decision of the U.S.S.R., in 1948, to suppress the use of any foreign 
language in scientific publications was a hard blow to those who — having painfully 
mastered French, German, and English — imagined that they then had access to the 
whole of the international scientific production. 

Astronomers are deeply indebted to Professor Otto Struve for the invaluable 
service he rendered to astronomy by the creation, in 1948, of the "Russian Section" 
of the Astronomical News Letters. This obviously met an urgent need. 

With the help of A. N. Vyssotsky and also that of S. Gaposchkin, he undertook 
the truly gigantic enterprise of making the vast astronomical literature published in 
the U.S.S.R. available as quickly as possible to the great majority of astronomers. 

Through the efforts -of Struve and his collaborators, the Astronomical News 
Letters (from No. 37 to No. 71) give a faithful picture of the astronomical production 
of the U.S.S.R. between 1948 and 1953. This is very clearly reflected by the Subject 
Index of the Astronomical News Letters, which is to be published in one of the next 
issues. 

Into the huge amount of summarizing work done in the past seven years Struve 
introduced a new method of bibliographical analysis, especially well-suited to his 
purpose ; these are its main features : 

(1) The notations used by the author in formulae, numerical tables, figures, etc., 
are translated and explained. This enables the reader, without any knowledge of 
Russian, to use the original paper for research purposes. 

(2) The reviewer expresses, besides discussing the physical content of the paper, his 
appreciation of ideas expressed by the author. He gives both a summary and a review. 

(3) All "delicate" topics, for instance, those involving the general and political 
outlook of Soviet scientists, receive an integral translation, as faithful as possible, in 
order to avoid any addition to the misunderstandings which at present make difficult 
the relations between the "East" and the "West". 



Bart J. Bok and V. Kourganoff 25 

Struve's method was so perfectly adapted to this aim and to present circumstances 
that, for the moment, there is no reason to introduce any change. It will be best to 
try to maintain the high standard of the past Astronomical News Letters, compen- 
sating the lack of universality of the present Editor by the division of the work between 
more specialists. 

To-day's organization of the Astronomical News Letters differs from the former 
one only in three points : 

(a) Kourganoff takes the place of Struve as Editor. 

(b) Extension of the former team, to include several European astronomers. 

(c) Some of the summaries will be written as previously in English, while others, 
according to the preference of the authors, will be written in French. 

Work on the summaries is centralized at Lille, where most of the manuscripts are 
typed and revised by the Editor, and then mailed to Harvard College Observatory, 
U.S.A. There Professor Bart J. Bok and Miss Olmsted take care of the printing 
and of the distribution. 

The modest financial help received by the Astronomical News Letters from the 
International Astronomical Union, the American Astronomical Society, and the 
French "Centre National de la Recherche Scientifique", is entirely absorbed by 
secretarial work, printing, and mailing expenses. 

For the present (March, 1954), American and Canadian observatories are paying 
$3 for five issues of the Astronomical News Letters, while all other institutes and 
observatories, on account of the difficulty of getting dollars, can receive, on applica- 
tion to the Editor, one copy of the Astronomical News Letter for their library. 
They can, of course, subscribe to more copies at the same rate as American and 
Canadian observatories. 

Though the Astronomical News Letter service has at present insufficient means for 
ambitious plans to be made, we expect to place the whole enterprise on a better 
financial basis to allow : 

(a) Simultaneous appearance of each issue in two languages: one entirely in 
English, and another entirely in French. 

(b) Complete translations of important papers and textbooks. 

(c) Extension to other languages than Russian. 

We hope that the description given above clearly indicates the spirit of the 
Astronomical News Letters : that of international friendship, service, and sincere 
political neutrality. 

V. K. 



Some Problems of International Co-operation in Geophysics 

J. M. Stagg 

Meteorological Office, Air Ministry, London* 

Summary 

To provide for new and rapidly developing subjects of great importance in geophysical science it is 
desirable that the constituent associations of the International Union of Geodesy and Geophysics should 
be modernized. This might also serve to keep the size of assemblies and the general administration of 
the Union within such bounds as can be effectively handled by an honorary secretary. The arrangements 
for administering grants-in-aid could also be simplified. An important aspect of international co-opera- 
tion is to maintain efficiently those permanent services of the Union whose function it is to collect and 
publish observational data and to provide standards, and individual countries should be encouraged to 
take responsibility for particular services. 



In geophysics as in astronomy international co-operation means more than the 
exchange of ideas and the encouragement of research : it includes the fostering of 
arrangements by which all countries interested in the physics of the Earth, its 
oceans, or its atmosphere, adopt common procedures of observation and make the 
observations available to all who wish to use them. In particular branches of study 
and between particular countries co-operation goes still farther ; but the stimulation 
of research by discussion and by exchange of publications, and the organization of the 
wherewithal to do it are the main objects of co-operation in geophysics on a world- 
wide scale. 

1. Present Organization 

Is the present machinery adequate for achieving these objects? Because of its 
special need for rapid interchange of data in standard form at fixed times each day 
over a network of stations in every country, meteorology is alone among the geo- 
physical sciences in having a separate organization, devoted primarily to the 
organizational side of the science. For geophysics as a group the principal medium is 
the International Union of Geodesy and Geophysics (U. G.G.I.) with a semi-autono- 
mous association for each of its seven main branches. So long as the boundaries 
between these branches remained fairly clear, the mechanism for ensuring inter- 
national collaboration worked well enough. But when new techniques and instru- 
ments of exploration, such as radar and high altitude rockets, and new lines of 
thought, like convection currents within the Earth, began to override the boundaries, 
some of the associations were found to have become too limited in their interests and 
responsibilities for adequate encouragement of the new fields of investigation. Ways 
of providing for these new branches had to be found. 

2. New and Borderline Subjects 

One solution has been sought in joint meetings of the associations most directly 
concerned. These serve the purpose of ventilating views, but they can be satisfactory 
in the larger sense only if the responsibility for developing the new subject and for 
organizing the necessary investigations lies specifically with one or other of the 



Lately General Secretary of the International Union of Geodesy and Geophysics (U.G.G.I.). 

26 



J. M. Stagg 27 

associations. Joint meetings cannot adequately provide the continuity of stimulus 
and support needed by new and rapidly expanding branches of the science. Nor does 
the formation of additional sections within existing associations or even the forma- 
tion of additional associations offer an adequate solution. Without forceful handling, 
sections having their own officers and their own programmes of activity are liable to 
become progressively more self-centred and self-contained, to the detriment of 
the parent association and its collaboration with other associations. And apart 
from the difficulty of finding accommodation at the Union's general assemblies for 
many associations and their committees in such close proximity as to preserve a 
community of spirit and opportunity for both formal and informal meetings, the 
present number of associations is about the limit that can satisfactorily be handled 
by an honorary secretariat, more especially since collaboration with other international 
bodies (e.g. UNESCO) has increased the amount of correspondence that must be 
dealt with centrally through the Union bureau. 

The most likely solution seems to lie in combining the smaller associations so as to 
make room for more vigorously expanding branches of the science ; or, less drastically, 
the alternative may lie in modernizing them so as to change the emphasis of their 
hitherto restricted interests and activities under new titles, and in widening their 
responsibilities accordingly. 

3. Permanent Services 
Another important problem concerns the maintenance of the permanent services 
which are so essential in geophysics, and of which U.G.G.I. is wholly or partly respon- 
sible for seven. Though most countries can contribute their quota of basic informa- 
tion about some aspect of geophysics, only a few have the facilities for applying the 
collected data to operational use or research on a world-wide scale. In meteorology 
the machinery required for this kind of collaboration is provided by World Meteoro- 
logical Organization, which, being a governmental body and also a specialized agency 
of United Nations with a full-time secretariat of about forty persons, has considerable 
resources through which it can function. The other geophysical sciences are less well 
provided. As membership of U.G.G.I. is by academic agreement, there is no legal 
obligation by member countries to implement the Union's resolutions or even to 
contribute to its funds. One indirect result is that, for a group of subjects each so 
dependent for development on measurements and observations made all over the 
world, geophysics has inadequate resources in bureaus for collecting, reducing, and 
publishing these basic data, and the bureaus that have been established lead an 
uncertain and precarious life. 

4. Affiliation with UNESCO 
As one possible way of making the permanent services more effective it has been 
proposed from time to time that membership of the Union should be by government 
convention. Since a change of this kind would undoubtedly alter the whole character 
of the Union as well as introduce serious difficulties for some member countries, it 
was not surprising that the proposal was allowed to lapse when UNESCO offered 
affiliation to the International Council of Scientific Unions and through I.C.S.U. to 
each of its unions, including U.G.G.I., particularly since the offer was made on the 
understanding that private (non-governmental) agreement would continue to be 
the basis of membership. As affiliation carried with it financial support, some of the 



28 Some problems of international co-operation in geophysics 

services sponsored by the Union (e.g. the compiling and publication of the Inter- 
national Seismological Summary) have been very materially helped, and others have 
been allowed to extend their important functions. Further, by relieving the Union 
and its associations of part of the increased cost of their publications and other 
expenses, the grants from UNESCO have permitted it to stimulate investigations 
which need assured support on a long-term basis. Financial assistance has also led 
to a rejuvenation of some of the older joint commissions and to the establishment 
of new joint commissions for the study of topics of mutual interest in borderline 
subjects. 

5. Grants-in-Aid 

With so much generous assistance on the credit side, other aspects of affiliation 
with UNESCO are introduced only because they form part of the whole picture. 
The acceptance of grants naturally puts the recipient unions under obligation to 
conform with UNESCO's procedures, and since the honorary officer of the union can 
seldom compete with the paid official of UNESCO, some of the unions have found it 
necessary to employ staff, and the others, including U.G.G.I., find it increasingly 
difficult to avoid following their example. This in itself is not evil, but it seems bound 
to lead to a tendency for secretariats to become anchored in particular countries 
and also to an extension of the bureaucratic element in the functioning of the 
unions. Furthermore the grants themselves tend to lead the associations to think in 
terms of budgets expanded to a degree which the normal resources of the union would 
have difficulty in maintaining should UNESCO policy change and the grants stop. 
To operate the scheme of grant allocation effectively would require each union to 
become more rigorously organized than is appropriate to the traditions of U. G.G.I. , 
and even then an element of unseemly scramble might remain. But perhaps the 
recent changes in procedure by which UNESCO gives a block grant to I.C.S.U. will 
improve these less welcome aspects, especially if I.C.S.U. is in turn allowed to give a 
block allocation to each of its adhering unions for approved but not rigidly defined 
schemes, the extent of the grant to be determined more by the essentially international 
character of the use to be made of it and less by criteria of present fashion and popular 
appeal. If to this were added some assurance of continuity of UNESCO policy so that 
plans could be made for terms of three or even five years, the unions and U.G.G.I. 
in particular could assure UNESCO that a more effective use could be made of the 
grants allotted. 

6. Joint Commissions 

A similar devolution of responsibility might with advantage be extended to the 
unions concerned with each of the joint commissions. At present these are set up 
by I.C.S.U. for the study of specific subjects which overlap the interests of two or 
more unions. I.C.S.U. provides grants for meetings and for the travelling and expense 
allowances of members : but because she cannot supervise the activities of all the 
joint commissions that have come into being — U.G.G.I. alone has representatives on 
six — I.C.S.U. nominates a mother union for each. Without question some of the 
commissions function admirably, but others develop a tendency to extend their 
interests, objects, and terms of responsibility farther than their terms of reference 
should permit. The weakness probably lies in the division of responsibility between 
I.C.S.U., who pays the piper, and the unions, who are required to call the tune. For 



V. Barocas 29 

it is not always easy for unions to consort together about the guidance to be given 
to a body whose activities are denned and paid for by I.C.S.U., and the mother 
union is often reluctant to seem to interfere, even though she may consider that the 
funds might be spread among other deserving projects. Perhaps the present arrange- 
ments for joint commissions would be improved if I.C.S.U. were to shed part of her 
financial responsibility on to the unions most intimately concerned with the work of 
each commission. 

7. International Obligations 
The best international co-operation is achieved when administration and finance 
either in the form of grants from bodies like UNESCO or even from central union 
funds are least obtrusive. As soon as payments are made to any commission or 
service, the others expect similar treatment and before long contributions have to be 
made to activities which individual countries had formerly helped to finance as part 
of their international obligations to science. In this way the machinery of administra- 
tion becomes more complex and expensive and resources intended for encouraging 
new projects and investigations are used up. The machinery becomes an end in 
itself. This process has not yet gone far in geophysics, though the signs are already 
there. Perhaps the best way to counteract it is to stimulate again the older-fashioned 
concept of co-operation as self-help in furthering common interests. This spirit in 
geophysics was at its best in the international polar years of 1882-83 and 1932-33, 
and is still potent between particular countries : perhaps it will be resuscitated on a 
wider basis in the forthcoming international geophysical year. 



Summary 



Some Educational Aspects of Astronomy 

V. Barocas 

Jeremiah Horroeks Observatory, Preston, England 



The interest in astronomy has much increased in recent years among people of all ages. The Preston 
Municipal Observatory has ventured the experiment of organizing the study of the subject in schools 
and by encouraging it also in evening classes, in lectures for the adult public, in local astronomical society 
work, and by active participation of advanced students at the observatory's activities. Some of the 
encouraging results obtained in this important field are discussed. 



Astronomy was in earlier centuries an accepted part of education, but has for a long 
time been either abandoned or relegated to a position of secondary importance by the 
general public. In recent years, however, a new interest in astronomy has arisen, 
and it is important that this interest should be fostered and directed along the right 
lines. This problem has already been recognized with regard to the interest shown by 
adults, but there is still a wide scope for work in this field among young people. 

The reasons for the increased interest shown by people of all ages are perhaps 
difficult to assess. Town-dwellers, who lived through the last war with its prolonged 



30 Some educational aspects of astronomy 

black-out and many night duties out of doors, became aware of the starry sky for the 
first time. Their curiosity was aroused and this led them to try to learn more about 
the heavenly bodies. During those years a number of simple books on star identifica- 
tion and similar subjects were published and sold very rapidly. 

The interest of the younger people has a completely different origin. It arises 
from science fiction films and "comics" dealing with space travel. The subject has 
proved very fascinating to young boys and girls and it is even reflected in their 
games. Unfortunately, the information obtained from these sources is very seldom 
accurate. As these interests seem to be increasing, it is more than ever desirable 
that young people should receive some sort of guidance to enable them at least to 
distinguish true scientific facts from fiction. Nor does this apply to young people 
alone. 

In the last few years a little time has been given to some aspects of astronomical 
problems in the syllabus of some grammar schools, where it is treated mainly from 
the point of view of applied mathematics. Little time can be afforded for it and it 
affects only a few of the pupils. Yet to-day, as astronomy expands in many new 
fields and new techniques and large instruments are being introduced, one of the 
main difficulties experienced in observatories all over the world is the lack of trained 
people to man fully these large instruments and to be able to deal with all the 
observations obtained. Apart, therefore, from research and routine work, a certain 
amount of educational work is desirable, if we wish to obtain a steady supply of new 
recruits for our science. This has already been realized in some schools fortunate 
enough to have either some kind of equipment, or what is more important, some 
enthusiastic teacher. The results have always been worth the amount of work and 
time spent by the teacher. Where small telescopes are available, a definite programme 
of work is often carried out, while in other schools, where there is no equipment, the 
astronomical societies, run by the pupils under the supervision of a master, keep 
interest alive. 

After leaving school, young people find an outlet for their interest in the many 
local astronomical societies, often founded in recent years, which generally have a 
junior section. The programme of these societies varies. Some are fortunate enough 
to have a telescope either bought or built by their members, enabling them to 
follow an observational programme; others limit their activities to lectures, dis- 
cussions, publication of a news sheet, and hope one day to be able to acquire a small 
observatory of their own. These societies are doing very good pioneering work, but 
many of them find that their funds are not sufficient to cover the many activities 
that they would like to undertake. A few societies are fortunate in having secured 
the interest of the local education authorities and receive some help from them under 
the Further Education Scheme. 

In most districts evening classes in astronomy are organized by the Workers' 
Educational Association, which provide a carefully graduated syllabus under the 
guidance of specialists. Finally we must not forget the Extra Mural Departments of 
the Universities and their valuable work. 

In this way a young person who is interested in astronomy has opportunities of 
extending his knowledge and furthering his studies. If he is unable to follow 
astronomy as a career, it may well become a very important hobby in his life. 
Amateur astronomers have done, and are still doing, very good work in this country. 
Very often they take a keen interest in certain aspects of astronomy which require 



V. Barocas 31 

patient and continuous observations, and which are generally not carried out by 
large professional observatories which are naturally occupied in more specialized 
work. Moreover, in a country with a varied and variable climate like Great Britain, 
the possibility of having a large number of observers scattered in all parts of the 
country is a great asset. The work done in this field by the B.A.A., the British 
Astronomical Association, is too widely known to need any description here. 

It is unfortunate that not more young people are aware of these opportunities. 
Schools can seldom afford as much as they would like to awaken interest. An 
observatory maintained by the local education authorities and available to the schools 
would appear to be an answer to this difficulty. There are only a few municipal 
observatories in the country but the educational work they do is of great value in the 
educational field. 

There is no doubt that a number of instruments are available in the country, 
which although not large enough for modern research work, could nevertheless be 
extremely useful for training purposes and for adult educational work. The main 
difficulty is the expense of acquiring and maintaining such equipment. An organized 
plan is needed for the erection and maintenance of a number of small observatories in 
the largest centres of the country. These observatories can do very varied work, as is 
illustrated by the activities of the municipal observatory, run by a local education 
committee as an institution for further education, and to which the writer is attached. 
This observatory has started an experimental educational programme aimed not 
only at increasing the interest in the subject of people who have already a general 
knowledge of astronomy, but also at giving sound scientific facts to people (child or 
adult) who are simply curious and who must otherwise rely for their information 
either on newspapers or on space travel films. 

The work has been concerned specifically with schools, the general public, the local 
astronomical society, and with more advanced students. The schools are visited 
periodically and simple lectures using film strips, slides, photographs, and models are 
given to young people (age thirteen to sixteen). The teachers also help by allowing 
their classes time to listen to the course of talks on astronomy for schools given by the 
B.B.C. After these talks, discussions usually follow and some of the points which 
may have presented difficulties are elaborated. When the class has a basic knowledge 
of elementary facts, it is then allowed to visit the observatory in the day-time so as 
to see the main equipment of an observatory and to receive a practical demonstration 
on how the instruments are used. Finally, the pupils visit the observatory, in small 
groups, at night, in order to see for themselves how celestial objects appear through 
a telescope. 

The experiment has shown that the young people are very interested and receptive. 
Some simple test-papers set at the end of the course have shown a general high 
standard. For the grammar school type of student, since one is talking to young people 
who have already obtained the G.C.E., and in some cases to pupils who have already 
gained admission to a University, the approach to the subject is different. The 
course generally consists of lectures dealing with particular aspects of astronomy. 
As the pupils have a knowledge of physics and mathematics, the lectures can be of a 
more advanced type. Here too the results obtained, judging by the standard of the 
questions asked, are very encouraging. Those students who show a definite interest 
in the subject are encouraged to come to the observatory to ask advice on further 
reading or on practical work. It is found that particularly among the boys there is a 



32 Some educational aspects of astronomy 

desire to do something themselves. Some observe simple phenomena with the naked 
eye or by means of binoculars, while one particular boy, who possesses a 3-in. refrac- 
tor, takes regular observations of the planets, making sketches of them and is now 
learning to take observations of occultations. This boy hopes to gain admission to a 
University and intends to make astronomy his career. 

For boys and girls who are older and who have already left school and are working, 
a series of lectures is arranged at the Day Continuation College which they attend. 
The attendance to these lectures is optional, but I have found that they are attended 
by all the students. 

The education of adults must be approached in a different way. The observatory 
is opened to the public one evening a week and, by arrangement, groups from 
societies and organizations can visit it on other nights. The number of people who 
visit the observatory during the year is very large. In these groups we often find 
people who express a wish to learn something more. To satisfy this demand evening 
classes are held where the more serious enquirers can learn some fundamentals of 
astronomy. They often join the local astronomical society and also the B.A.A. 
Apart from this, lectures for the general public, given by eminent astronomers, are 
organized every year. 

Three years ago an astronomical society was formed in the town. The members 
meet regularly for lectures and they visit the observatory periodically, where they 
can obtain information and can consult periodicals and publications of other observa- 
tories. A few members have purchased or constructed small telescopes, and an 
observational programme is being prepared. 

Finally, the observatory is also available for more advanced students. Some are 
people who are preparing for an external degree of London University and who, 
having chosen astronomy as a subsidiary subject, require practical knowledge of the 
use of astronomical instruments. Others are preparing for examinations in surveying, 
and they too require a knowledge of field astronomy and practical observations. 

On the whole it is felt that the new experiment has been very successful. It 
provides the schools and the residents of the town and district with something that 
was lacking before. I feel that although the work may not be so immediately reward- 
ing as research work, it does perform a necessary service to the science of astronomy. 
It helps to foster an interest in astronomy, provides a starting point for those who 
wish to acquire some elementary knowledge which can later be developed, and enables 
them to appreciate the work and the problems that face the astronomer to-day. 
It is interesting to note that the general reaction of adult visitors to our observatory 
is one of envy that young people to-day have this opportunity of learning about 
astronomy which was lacking in their young days. 

The introduction of this subject to young people brings a realization of new and 
wider horizons. It teaches and encourages them to think for themselves instead of 
accepting passively the evidence their eyes see about them, and it provides, especially 
for the more thoughtful child, a source of inspiration that is so often lacking in this 
materialistic age. It is, indeed, an introduction to the power of the human mind 
and lays the foundations for a fuller life bringing with it the pleasure and appreciation 
that comes from reading and sharing the experiences of our great scientists, who 
through centuries of patient work and their interpretation of the observations have 
gradually built up the knowledge of the Universe that we have to-day. 



Recollections of Seventy Years of Scientific Work 

J. EVERSHED 

Ewhurst, Surrey, England 
Summary 

This article records some of the researches in solar physics undertaken during my life. These have 
included the following: prominence observations covering sixteen years; spectroheliograph work with a 
direct vision prism; experiments on the radiation of heated gases; eclipse expeditions in 1898 and 1900; 
studies of the relation between emission and absorption spectra, and of the continuous spectrum of 
hydrogen. In India (1906-1923) my work included: the discovery of radial motion and the estimate 
of pressure in sunspots; a study of the exceptional observing conditions in Kashmir; measurements of 
the red-shift in connection with Einstein's prediction; great magnetic storms during flares; the motion 
in the tail of Hailey's comet and its transit over the Sun ; observations of novae. At Ewhurst (1923-1 954) 
I was concerned with : high -dispersion work with liquid prisms ; the study of the Zeeman effect and the 
discovery of a particularly sensitive line for its determination; measurements of minute line-shifts 
due to horizontal motions ; and the decrease of wavelength of solar lines. Reference is made to scientific 
men I have met and to whom I am indebted for interest and encouragement. 



1. Early Memories 
My interest in astronomy began over eighty years ago, when, as a small boy, there 
happened a partial eclipse of the Sun, and I walked, or ran most of the way, from 
Gomshall to Shere to see it in the doctor's telescope. An added excitement and one 
which was destined to occupy my thoughts for the rest of my life was a spot on the 
Sun's disk. 

I can remember also, at the age of six, seeing in an illustrated paper a picture of 
German shells falling in the streets of Paris, and it was during this siege by the 
Germans that Janssen, the French astronomer, escaped by balloon from the 
besieged city to observe a total eclipse of the Sun. Was it not on this occasion, 
when astonished by the brilliance of the red line of hydrogen he determined to look 
for it again in full sunshine ? This was the beginning of the daily spectroscopic 
observation of prominences without waiting for an eclipse. 

About the year 1875, my eldest brother, who was a student at the School of Mines 
in London, became acquainted with Raphael Meldola, F.R.S., who later became a 
friend of the family and brought us news of the scientific world. He was a friend of 
Charles Darwin, with whom I had the great honour of speaking when Meldola 
took me to the Royal Society. He also introduced me to the naturalist, Alfred 
Russell Wallace. My interest in evolution and in natural history was thereby 
greatly stimulated. 

In later years I was impressed by Lockyer's articles in Nature about solar 
prominences, and desiring to see these marvels myself, I made a spectroscope with a 
tiny prism of 1 cm aperture and a pair of lenses taken from a disused opera-glass. 
This revealed the solar spectrum with its wonderful array of dark slit images, but was 
useless for the prominences. Later I constructed a spectroscope with a train of small 
prisms after the pattern of Lockyer's instrument. With this attached to a 3-in. 
telescope I got beautiful views of the prominences ; and so began a long series of 
observations and records with a view to determining their distribution in solar 
latitude and their connection with sunspots. 

The discovery by Hale in 1891 of the reversals of the lines H and K in scattered 
regions over the surface of the Sun and in the prominences, and his subsequent work 
4 33 



;m 



KoculltKitioiiH of aovonty years of soientifio work 



at the Kenwood Observatory with the beautiful instrument he designed for recording 
these phenomena, led me to abandon some attempts I had made to photograph 
prominences and disk markings in the hydrogen line (i and to see what. T could do 
with the calcium lines. Thin led to sonic interesting correspondence between Kenwood 
and my home at Kenley. 

In the year 18fW I became acquainted with Mr. Cowper Hanvard, F.R.A.S., and 
lie introduced me to Half*;, who was on a visit to England. Ran yard died in 
December, I.SD4, and left to me his astronomical equipment which included an 
IS-in. reflecting telescope, and a smalt speetroheliograph which he lent to Hale for 
use on El na in an attempt to photograph the corona in full daylight. This instrument 
had the disadvantage of giving distorted spectral images of the Sun, owing to the 
action of the prisms in giving curved spectrum lines, I erected the telescope and its 
dome at Keiilex . bin being dissatisfied with the distorted images given by the speetro- 
heliograph, 1 designed an instrument using a large direct vision prism giving straight 
lines and undistorted solar images. Tins type of speetroheliograph was adopted and 
successfully used at the Torlosa Observatory in Spain (Fig. 1). 




l""ig. 1. My first spectra - 
heliograph of ISM {with 
cliivrt vision prism ; attached 
to the 18 in. telescupo). 



In the year 189a it was generally believed that the radiation of gaseous elements 
could not be produced bv heat alone, but only by electrical or chemical stimulation 
could their characteristic line spectra be produced, and this, of course, applied to the 
spectrum of the Sun. The researches of Pkinoshkim, Pakchkn. and Smith kt.lk in this 
connection induced me to undertake some interesting experiments on the radiation 
of heated gases, 1 was able to show first that the coloured vapours of iodine and 
bromine heated to the temperature of a red heat glowed with a continuous spectrum, 
at the same time giving a discontinuous absorption sped rum by transmitted light. 



J. EVERSHED 35 

Other coloured vapours, including chlorine, also gave a continuous spectrum. I 
was also able to show that vaporized sodium could be made to emit its characteristic 
D radiation by heat alone under conditions where there could be no action other 
than heat. These results were published in The Philosophical Magazine (Evershed 
1895). 

2. Eclipse Expeditions 
In 1898 I joined an eclipse expedition to India organized by the British Astronomical 
Association. I hoped to get photographs of the spectrum of the reversing layer to 
study the relation of the emission spectrum to the Fraunhofer dark line spectrum. 
I obtained good spectra extending far into the ultra-violet. A new feature was 
shown in the hydrogen spectrum of prominences, consisting of a continuous spectrum 
beginning at the limit of the hydrogen series of lines at A3646 and extending to the 
end of the plate. This proved to be the counterpart of the continuous absorption 
spectrum discovered by Sir William Hitggins in stars having very strong hydrogen 
lines. 

The results of measures of wavelength and intensity in the flash spectra at this 
eclipse were published in The Philosophical Transactions of the Royal Society (Ever- 
shed, 1902) and also those of the eclipse of 28th May, 1900 (Evershed, 1903). The 
general result of these two eclipses showed that the flash spectrum represents the 
higher, more diffused portion of the gases which by their absorption give the Fraun- 
hofer dark line spectrum, and that the flash spectrum is the same in all solar latitudes. 
Previously, it was considered that the flash and the Fraunhofer spectra were separate 
and distinct phenomena. 

At the 1900 eclipse in Algeria a station was chosen near the limit of totality instead 
of, as is usual, on the central line of eclipse. Near the limit the duration of the 
"flash" is increased from about 2 sec. to 30 sec. or more. Actually, owing to irregu- 
larities in the contour of the Moon the eclipse was not quite total, a small point of the 
Sun's disk remaining visible at mid-eclipse. 

Some Arabs at my station were arguing about whether the eclipse was total or not ; 
the berger stated that some little piece of "el Simpsh" remained, as much, he said, as 
a "garro" (cigarette), but this was contradicted by some men who were hoeing maize 
500 metres north-east of my camp, these all declared that the whole Sun was obscured 
for a moment, as it no doubt was. 

The results obtained from the spectra of these eclipses led to an interesting corres- 
pondence with Sir William Huggins, then President of the Royal Society. It was, I 
think, in the year 1897 that I first had the privilege of visiting Sir William at Tulse 
Hill, where he showed me his instruments and explained how he had discovered the 
gaseous nature of the Orion nebula. In the year 1905 it was largely through his 
influence that the India Office offered me the post of assistant to Mr. Michie-Smith, 
director of the Kodaikanal Observatory. 

I must here refer to the very helpful advice given me by the late Professor H. H. 
Turner, who arranged for me to travel to India via the United States and Japan, 
and provided me with introductions to the leading American astronomers. 

After interesting and instructive visits to Harvard and Yerkes Observatories, my 
wife and I arrived at Mount Wilson, where we stayed for some time, and I had 
excellent opportunities for studying their instruments and methods and the work 
being carried out under the inspiring direction of Professor Hale, 



30 l{(H:olli!i:l,ion,H of seventy years of* [scientific mirk 

3. India, 1906-23 

My early work at Kodaikanal was largely concerned with the Cambridge speetro- 
heliograpli which I brought into working order early in 1907; and ho began the long 
series of photographs of the Sun's disk ;md Mie prominences. 

The observatory possessed an excellent grating of about 70, 000 lines, and with this 
I constructed a high dispersion spectrograph. 1 used this at first to estimate the 
pressure ill the reversing layer and in sunspots, and to find out whether there was any 
motion of ascent or descent in sunspots, The "first result siiowed that there could be 
no appreciable pressure in (he reversing layer or in sunspots. An excellent oppor- 
tunity presented itself for this work on f>th January, 1900, with two large spots on 
the Sun, and excellent definition after heavy rainstorms. The spectra revealed a 
curious twist in the lines crossing the spots which I at once thought must indicate a 
rotation of the gases, as required by Hat.e'k recent discovery of strong magnetic 




Fig. "1. An t?arly stage of u 
spectacular solar eruption, re- 
corded in Kashmir on 2<>th May, 



helds in spots, but it soon turned out from spectra taken with the slit placed at 
different angles across a spot that the displacement of the lines, if attributed to 
motion, could only be doe to a radial accelerating motion outward from the centre 
of the umbra. Later photographs of the calcium lines H and K and the hydrogen 
line k, revealed a contrary or inward motion at the higher levels represented by these 
lines (KvfjRkhkd, 1010a}. 

In January, ION, Mr. Miohie-Sm.i:tjh retired and I became director of the 
observatory, 

My late wife was greatly interested in the prominences I had recorded at Kenley 
and in those we had studied together with the fine equipment at Kodaikanal. She 
was much occupied at this time in writing her important w r ork on "Dante and the 
Early Astronomers". She nevertheless was able to find time to study in detail the 
prominences recorded in the years 1890-1914 inclusive. The results were published 
In a Memoir of the Kodaikanal Observatory. 



J. EVTORSHBP 



81 



In 1914 Hale informed me that he was getting very interesting spcctroheHograms 
using the red hydrogen line a, and now that red sensitive plates were available I 
attached to the Cambridge instrument a second spectroheJiograph, using a grating 
and special arrangements for getting disk photographs in hydrogen light, uti lizing 
the very perfect movement of the Cambridge instrument. Henceforward, speetro- 
heliograms were taken daily in both calcium and hydrogen light. Notable examples 
were obtained on the 10th August, 1917. Several spectacular eruptions wore recorded, 
a very interesting one on 26th May, 1016, in which the speed of recession of the 
flying fragments was measured. This was photographed at Kodaikanal and in 
Kashmir (Kvgiistted, 1914); (Fig. 2). 

Owing to the excellent conditions prevailing in the Valley of Kashmir a temporary 
observatory was established there hi 1915, where 1 installed a large speotroheliograph 




K«£. :i. Temporary observatory established in LA IS in tlio Valley of Kashmir 



(Fig. 8) and co-operated with Dr. T. RoYDS who had charge, in my absence, of the 
Kodaikanal Observatory. 

The Valley of Kashmir is a level plain containing a river and much wet cultivation 
of lice. It is 5,000 ft above sea level and is completely surrounded by high mountains. 
Under these conditions the solar definition is extremely good at all times of the day. 
and an like most high level stations it is best near noon and in hot summer weather. 
After experiencing these remarkable conditions, observations were made of solar 
definition in various localities, including one on an ocean liner in tropical waters far 
from land, The best solar definition was found in low level plains near the sea, or on 
small islands surrounded by extensive sheets of water. At sea, the definition appeared 
to be perfect so far as could be judged, with a moving solar image. 

A large seetion of our work at Kodaikanal was devoted to the measurement of the 



38 



Rouollootions of seventy yeara of scientific wori 
N 




W 



Fig. 4. Enlargement of a calcium 
spot-troholkigrtnn, obtained on 24th 
September, 1900. This large sun- 
rtjint wtw responsible foi* the great 
1 1 uitftictie storm on the 25th, which 
npsf-t tlic Indian telegraph system 



J I I L 



12 3 4 
Minutes of arc 



minute shifts of the .solar lines towards the red. especially the iron lines, when com- 
pared with spectra obtained with the electric are. In general, in light from the 
centre of the Suns disk, the shifts were found to agree more or less with the shift 
predicted by Kikstein, but at the Sun's limb the shift was greater, almost by a 
factor of two. and this excess remains an unsolved problem. 

The integrated light of the whole disk of the Sun also gives spectrum lines shifted 
to red in excess of the predicted shift, and it occurred to me to put the planet Venus 
into my optical train in order to set; whether or not this excess shift would appear 
in light reflected from the far aide of the Sun. Yet here again when the planet was 
neariug superior conjunction the line shifts were unaltered. 

Our time in India was characterized by some interesting and, indeed, exciting 
events: 

(1) A great, spot in Sept ember, ItlOll (Fig. 4), gave a spectrum in which the Zeeman 
splitting of the iron lines in the region near H and K indicated a magnetic effect of 
about 10,0(10 gauss during what would now be called a "flare" (Evtcrshtcd, 1910b), 
The accompanying magnetic storm upset the Indian telegraph system, for which 
we appeared to be held responsible by the Director of Telegraphs ! 

(2) On the morning of 17th January, 10 10, when we were starting the daily series 
of speetroheliograins, there appeared a great sun-grazing comet shining in the blue 
sky with a brilliance almost like a. detached piece of the Sun itself. In April and May 
of this same year H alley's comet gave ns a memorable display, with its tail extend- 
ing for .100° far up towards the zenith. Many interesting photographs were obtained 



J. EVERSHED 39 

and these revealed an accelerating outward movement of the tail. It appeared as 
though the nucleus had thrown off an entire tail and was forming a new one. 

On May 19th the nucleus was computed to be in transit over the Sun. We attempted 
to photograph this event with the spectroheliograph, using the ultra-violet light of 
cyanogen which is a specially prominent radiation of the head of comets, but no 
trace of the comet could be seen in the photograph, nor could it be seen on the Sun 
in an ordinary telescope. At this time the tail must have swept over the Earth and a 
sensation was caused in the newspapers by the suggestion of a poisonous gas entering 
our atmosphere. 

(3) On 9th June, 1918, a brilliant new star appeared and the resources of the 
observatory had to be mobilized at very short notice. In this I had the assistance of 
the late Mr. R. J. Pocock of the Nizamiah Observatory, who was visiting Kodaikanal 
at the time, and it was due to his very resourceful help that we were able to get all the 
arrangements perfected on 12th June. It is interesting to recall some of the results 
obtained. The hydrogen emission lines assumed extraordinary forms, Ha appeared 
as a block of brilliant red light, the /? and y lines were more extended blocks of blue 
and violet, well defined at their edges. Interpreting these wide bands as due to 
motion in the line of sight, an expansion of the hydrogen was indicated with velocities 
up to 1 580 km per sec. On the violet sides of these bands two absorption lines appeared, 
one on the edge of the bright bands ; these indicated velocities of the absorbing 
hydrogen on 12th June of 1586 and 2193 km per sec. respectively. After 13th June 
the more refrangible line died out, but the other remained, indicating by its decreasing 
wavelengths on successive days an accelerating outward motion. This motion 
outward from the star was shared also by the iron vapour (Evershed, 1919). 

Our somewhat isolated life at Kodaikanal was relieved occasionally by visitors 
from home or other parts of India. The Director of Indian Observatories, Sir 
Gilbert Walker, came on his official visits from Simla once every year with Lady 
Walker, a most welcome interlude. 

4. Ewhurst, 1923-55 

On retiring from the Indian Meteorological Service in 1923, and establishing a 
private observatory near Ewhurst, I can recall again the interest and encouragement 
of H. H. Turner, F.R.S., and of his friend, Sir Charles Parsons. Sir Charles 
supplied a large block of optical glass which was essential, when made into a 6 in. 
prism, for much of the work undertaken. This included daily spectroheliograms in 
hydrogen light, and other spectrograms which confirmed and extended to the promi- 
nence region the very remarkable fact that the angular rotation of the Sun increases 
with the height above the photosphere (Evershed, 1935). 

My early experiments with a hollow prism filled with ethyl cinnamate revealed great 
possibilities for high dispersion work if certain formidable difficulties could be over- 
come. Here I was fortunate to have the co-operation and very skilled work of 
Mr. F. J. Hargreaves, who not only perfected the first prism used (1938), but 
designed and made larger prisms giving very high resolving power. These liquid 
prisms have to their credit many beautiful spectrum photographs which have revealed 
some interesting facts ; in particular in the magnetic field of sunspots (Evershed, 
1939a and 1944), the significant fact was disclosed that lines which are Zeeman 
triplets over the umbrae are doublets in the penumbrae. They also made possible the 
detection of minute shifts indicating movements parallel to the Sun's surface over 



40 



Hi'colliH-.Lions of seventy years of scientific! work 



25*//- . / 8 13 &q*j $„marA&*» 


1 


AlilkJLj 




| 




1 
1 


(r tit? SS 


IS >/*.•$? <p = Q* 


a #•/£•/& 



Fig. 5. Sprrtra showing the D -lines 
of HuUhiiii in tin- Sun mitt in a 
viHuuni lube, uiiii illustrating tin- 
perfection of the liipiid prium 
(I en transmissions through n single 

prism). ()ii/*j/j; photograph ttik<i 

jr»rli November. 11)37, at 8 1 ' 28"' 
(!MT. Centre mul bottom: these two 
narrow s|H'etm. b mul «, were photu- 
zraphed on the same day (at about 
I t 1 ' HJ m ) at the west anil east limbs 

ill tin- Mdlat' i,'(|iia!<ir. ll'H]KH-t IVfly. 

They shuw also llii" bright, lines 

[>, ami D a given by n vacuum tulie 

(■onlmtiiiit: siiiliuin 



wide areas of tJie photosphere (1033), and gave evidence of decreasing wavelengths 
during recent years, of iron and nickel lines in spectral regions when- atmospheric 
lines of oxygen mid water vapour formed lixed standards of reference (Rvkiikiieii, 
1049); see Fig. 5. 

Throughout my time in India and at Ewhurst in later years I have owed much to 
the friendship of the late Professor H. F. Nbwall, F.R.S., and I recall with great 
pleasure (lie many visits to his beautiful home at Madingley Rise. Cambridge. In 
1933 n memorable event was a visit from Nbwall and Bale to Ewhurst, when 
SALE brought me Ins original Spectra whieh were thought to give evidence of the 
Sun's general magnetic field. These were measured by the positive on negative 
method with the final result that any minute shifts that could he detected indicated 
Doppler and not Zeeman effects (Kvkrshed, 1939b). 

Finally, it is interesting to recall that I began observing the Sun with a tiny prism 
of half-inch aperture, and have ended with giant solid and liquid prisms of five to 
six inches effective aperture. 



KvKItMIKl), J. 



RjKVWUUKHM 




iS'irj 


/'/-;/. Mag., (May), f>. 400, 


1!I0L> 


I'hil. Tratu^ 197, 381. 


LAOS 


i'tiiL Tram., 201, 487. 


L910a 


ftfJV., 70, 229. 


1910b 


Kodaikanal Obs, Hull.. X->- 22 


1914 


Kadaikanal ah.*. Butt., No. A'i 


1919 


.1/..V..79, 488. 


I :w:j 


M.S.. 94, 9ft. 


i !>:i:> 


M.S.. 95, 609. 


1938 


MM,, 99, 127. 


1939a 


MJf„ 99, ^17. 


1939b 


1/..V..99, 488. 


1044 


iHwrrufitn/, 65, l!>n. 


1948 


.U..V.. 109, 698. 



Summary 



Astronomical Recollections 

Max Born 

Department of Mathematical Physics, University of Edinburgh* 



Reminiscences of the author's experiences, fifty years ago, as a student of astronomy at two German 
universities, Breslau and Gottingen, and of his teachers, Franz and Schwarzschild. 



I am not an astronomer, nor have I done any work in physics applicable to astronomy. 
Yet I cannot resist the wish to be included amongst those who offer their congratula- 
tions to Professor Stratton by an article in this volume. There was a time in my 
life when I was very near to devoting myself to the celestial science ; but I failed. 
May I offer, as a substitute for a more serious contribution, the story of my wrangling 
with astronomy and some recollections of remarkable astronomers who were my 
teachers. 

I have to begin with Professor Franz, the director of the observatory of my home 
city, Breslau. My father, who died just before I finished school, had left me the 
advice to attend lectures on various subjects before choosing a definite study for a 
profession. In Germany of that period this was possible because of the complete 
"academic freedom" at the university. 

There was in most subjects no strict syllabus, no supervision of attendance, no 
examinations except the final ones. Every student could select the lectures he liked 
best ; it was his own responsibility to build up a body of knowledge sufficient for the 
final examinations which were either for a professional certificate or for a doctor's 
degree, or both. Thus I made up a rather mixed programme for my first year, 
including physics, chemistry, zoology, general philosophy and logic, mathematics 
and astronomy. At school I had never been very good nor interested in mathematics, 
but at the university the only lectures which I really enjoyed were the mathematical 
and astronomical ones. The greatest disappointment were the philosophical courses ; 
there we heard a lot about the rules of rational thinking, the paradoxes of space, 
time, substance, cause, the structure of the universe, and infinity. Yet it seemed to 
me an awful muddle. Now the same concepts appeared also in the mathematical 
and astronomical lectures, but instead of being veiled in a mist of paradox they were 
formulated in a clear way according to the case. For that was the important dis- 
covery I then made : that all the high-sounding words connected with the concept of 
infinity mean nothing unless applied in a definite system of ideas to a definite problem. 

Astronomy was attractive in another way. There the problems of cosmology are 
related to the infinity of the physical universe. But little about these great questions 
was mentioned in the elementary lectures of our Professor Franz. What we had to 
learn was the careful handling of instruments, correct reading of scales, elimination 
of errors of observation and precise numerical calculations — all the paraphernalia 
of the measuring scientist. It was a rigorous school of precision, and I enjoyed it. 
It gave one the feeling of standing on solid ground. Yet actually this feeling was not 
quite justified by facts. The Breslau Observatory was not on solid ground, but on 

* Now at Bad Pyrmont, Germany. 

41 



42 Astronomical recollections 

the top of the high and steep roof of the lovely university building, in a kind of roof 
pavilion, decorated with fantastic baroque ornaments and statues of saints and angels. 
The main instrument was a meridian circle, which a hundred years ago had been 
used by the great Bbssel ; although it was placed on a solid pillar standing on the 
foundations and rising straight through the whole building, it was not free from 
vibrations produced by the gales blowing from the Polish steppes. The whole outfit 
of this observatory was old-fashioned and more romantic than efficient. There were 
several old telescopes from Wallenstein's time, like those Kepler may have used. 
We had no electric chronograph but had to learn to observe the stars crossing the 
threads in the field of vision by counting the beats of a big clock and estimating the 
tenths of a second. It was a very good school of observation, and it had the additional 
attraction of an old and romantic craft. 

I remember many an icy winter's night spent there in the little roof pavilion. We 
were only three students in astronomy, and we took the observations alternately. 
When my turn was finished I enjoyed looking down on the endless expanse of 
snow-covered, gabled roofs of the ancient city, the silhouettes against the starry 
sky of the massive towers of the churches around the market place and of the 
Cathedral further away beyond the river. There on the narrow balcony amongst the 
stucco saints and old-fashioned telescopes, one felt like an adept of Dr. Faustus and 
would not have wondered if Mephistopheles had appeared behind the next pillar. 
However, it was only old Professor Franz who came up the steps to look after his 
three students — he had not had so many for a long time — and who carried with 
him the soberness of the exact scientist, checking our results and criticizing our 
endeavours with mild and friendly irony. 

These, our results, I rather think were not very reliable ; it was not so much our 
fault as that of the exalted but exposed position of the observatory. Professor 
Franz himself, therefore, abstained from doing research, which needed exact 
measurements, and restricted himself to descriptive work, a thorough study of the 
moon's surface which he knew better than the geography of our own planet. He 
made strenuous efforts, however, to obtain a modern observatory but never succeeded. 
During my student time there were great hopes. The firm Carl Zeiss, Jena, had sent 
a set of modern instruments to the World's Fair at Chicago. After the end of the 
show these were purchased by the Prussian State for its university observatories. 
Breslau obtained an excellent meridian instrument and a big parallactic telescope ; 
yet no proper building was granted, and the meridian circle was installed in a 
wooden cabin on a narrow island of the Oder River, just opposite the university 
building. This island was in fact an artificial dam between the river and a lock 
through which many barges used to pass. The time service for the province of 
Silesia, which had been practised for scores of years with the help of the old Bessel 
circle, was transferred to the new Zeiss instruments, but the results remained highly 
unsatisfactory. Eventually we discovered a correlation between the strange irregu- 
larities of the time observations with the changing level of the water in the lock ; 
the island suffered small displacements through the water pressure. Professor 
Franz's hopes of a more efficient observatory had broken down again. 

We youngsters took this disappointment rather as a funny incident. It did not 
diminish the fascination which astronomy exerted on my mind. This fascination was, 
however, shattered by the horrors of computation. Franz gave us a lecture on the 
determination of planetary orbits, connected with a practical course where we had 



Max Born 43 

to learn the technique of computing, filling in endless columns of seven decimal 
logarithms of trigonometric functions according to traditional forms. I knew from 
school that I was bad at numerical work, but I tried hard to improve. It was in 
vain, there was always a mistake somewhere in my figures, and my results differed 
from those of the class mates. I was teased by them, but that made it worse. I do 
not think that I ever finished an orbit or an ephemeris, and then I gave up — not 
only this calculating business but the whole idea of becoming an astronomer. If I 
had known at that time that there was in existence another kind of astronomy 
which did not consider the prediction of planetary positions as the ultimate aim, but 
studied the physical structure of the universe with all the powerful instruments and 
concepts of modern physics, my decision might have been different. But I came in 
contact with astrophysics only some years later, when it was too late to change my 
plans. 

At that period German students used to move from one university to another, 
from different motives. Sometimes they were attracted by a celebrated professor or a 
well-equipped laboratory, in other cases by the amenities and beauties of a city, by 
its museums, concerts, theatres, or by winter sport, by carnival and gay life in 
general. Thus I spent two summer semesters in Heidelberg and Zurich, returning 
during the winter to the home university. The observatory of Heidelberg was on 
the Konigstuhl, a considerable, wooded hill, where the astronomers lived a secluded 
life remote from the ordinary crowd. I had then definitely changed over to physics, 
and not even the celebrated name of Wolf, the professor who has discovered more 
planetoids than anybody else, deflected me from my purpose. 

The observatory in Zurich was more accessible, and the name of the professor was 
Wolfer, which could be interpreted as a comparative to Wolf. But even that 
did not attract me. 

The following summer I went to Gottingen for the rest of my student time. There 
Karl Schwarzschild was director of the famous observatory which had been for 
many years under the great Gauss. Schwarzschild was the youngest professor of 
the university, about thirty years of age ; a small man with dark hair and a moustache, 
sparkling eyes and an unforgettable smile. I joined his astrophysical seminar and 
was for the first time introduced to the modern aspect of astronomy. We discussed 
the atmosphere of planets, and I had to give an account of the loss of gas through 
diffusion against gravity into interstellar space. Thus I was driven to a careful 
study of the kinetic theory of gases which then, in 1904, was not a regular part of the 
syllabus in physics. But this is not the only subject, which I first learned through 
Schwarzschild's teaching. His was a versatile, all-embracing mind, and astronomy 
proper only one field of many in which he was interested. About this time he 
published deep investigations on electro -dynamics, in particular, on the variational 
principle from which Lorentz's equations for the field of an electron and for its 
motion could be derived. In the following year (1905) there appeared the first of his 
great articles on the aberrations of optical instruments ; these are, in my opinion, 
classical investigations, unsurpassed in clarity and rigour, by later work. I have 
presented this method in my book Optik (Springer, 1932), and it is again to be the 
backbone of a modernized version which will appear soon as an English book on 
optics (in collaboration with E. Wolf*). Schwarzschild applied his aberration 

» Pergamon Press, London. To be published in 1956. 



44 Astronomical recollections 

formulae to the actual construction of new types of optical systems ; but I am not 
competent to speak about this part of his activities. Nor can I discuss his astro- 
nomical work, experimental or theoretical. Personally he was a most charming man, 
always cheerful, amusing, slightly sarcastic, but kind and helpful. He once saved me 
from an awkward situation. I had intended to take geometry as one of my subjects 
in the oral examinations for the doctor's degree, but was not attracted by the 
lectures of Felix Klein, the famous mathematician, and attended somewhat 
irregularly. This fact did not escape Klein's observation and he showed me his 
displeasure. A disaster at the orals, only six months ahead, seemed to be impending. 
But Schwarzschild said that half a year was ample time to learn the whole of 
astronomy. He gave me some books to read and tutored me a little, in exchange for 
my training him in tennis. When the examination came his first question was : 
"What do you do when you see a falling star?" Whereupon I answered at once: 
"I have a wish" — according to an old German superstition that such a wish is always 
fulfilled. He remained quite serious and continued : "Yes, and what do you do then ? " 
Whereupon I gave the expected answer : "I would look at my watch, remember the 
time, constellation of appearance, direction of motion, range, etc., go home and work 
out a crude orbit". Which led to celestial mechanics and to a satisfactory pass. 
Schwaezschild differed from the ordinary type of the dignified, bearded German 
scholar of that time ; not only in appearance, but also in his mental structure, which 
was thoroughly modern, cheerful, active, open to all problems of the day. Still he 
had his hours of professorial absent-mindedness. There was a "Stammtisch", a 
certain table in a restaurant where a group of young professors and lecturers used to 
meet for lunch. Schwarzschild was one of them until his marriage. A few weeks 
after the wedding he was again at his accustomed place at the lunch table and plunged 
in his usual way into a lively discussion about some scientific problem, until one of 
the men asked him: "Now, Schwarzschild, how do you like married life?" He 
blushed, jumped up, said: "Married life — oh, I have quite forgotten — ", got his 
hat and ran away. But I think this kind of behaviour was not typical of him. He 
always knew what he was doing. His life was short, his achievements amazing, his 
success great — his end tragic. When the great war of 1914-18 broke out he was 
employed as a mathematical expert in ballistics and attached to the staff of one of 
the armies on the Eastern front. There, in Russia, he contracted some rare infectious 
disease. It was said that he refused to be sent home, until it was too late. On his 
way home, he visited me in my military office in Berlin ; he was still cheerful, but he 
looked terribly ill. Soon afterwards he died. Now his son, Martin, keeps up the 
astronomical tradition, thus founding another one of those hereditary lines of 
astronomers, the Herschels, the Struves, and so on. 

I have met many other distinguished astronomers and been intimate with some of 
them ; but as most of them are still wandering on this globe, I had better refrain 
from telling stories about them. 

May I conclude by wishing Professor Stratton a happy future and by adding 
the request that he too may present us with some recollections of astronomical 
personalities out of his long experience. 



SECTION 2 



HISTORY 

AND 

PHILOSOPHY 



"Glaubt ihr derm, dass die Wissenschaften entstanden und 
gross geworden waren, wenn ihnen nicht die Zauberer, 
Alchimisten, Astrologen und Hexen vorangelaufen 
waren als die, welche erst Durst, Hunger und 
Wohlgeschmack an verborgenen und verbotenen Machten 
schaffen mussten?" 

Fbiedbich Nietzsche, Frohliehe Wissenschafl, IV, 1882. 



The Egyptian "Decans" 

O. Neugebauer 

Brown University, Providence, R.I., U.S.A. 

Summary 

It is shown that the 10° sections of the ecliptic, called decans by the Greeks, were originally constellations 
rising heliacally 10 days apart, and invisible for 70 days. Such stars belong to a zone south of the ecliptic 
and include Sirius and Orion. The use of the decans for time measurement at night leads to a twelve- 
division of the period of complete darkness. From this is eventually derived the twenty -four division 
of day and night. 



1. Three different systems of astronomical reference were independently developed 
in early antiquity: the "zodiac" in Mesopotamia, the "lunar mansions" in India, 
and the "decans" in Egypt. The first system alone has survived to the present day 
because it was the only system which at an early date (probably in the fifth century 
B.C.) was associated with an accurate numerical scheme, the 360-division of the 
ecliptic. The lunar mansions, i.e. the twenty-seven or twenty-eight places occupied 
by the Moon during one sidereal rotation, were later absorbed into the zodiacal 
system which the Hindus adopted through Greek astronomy and astrology. With 
Islamic astronomy the mansions returned to the west but mainly as an astrological 
concept. A similar fate befell the decans. When Egypt became part of the Hellen- 
istic world the zodiacal signs soon show a division into three decans of 10° each. As 
"drekkana" they appear again prominently in Indian astrology, and return in 
oriental disguise to the west, forming an important element in the iconography of 
the late Middle Ages and the Renaissance. 

2. We shall be concerned not with these wanderings of early astronomical concepts 
but with the much discussed problem of the localization of the decans. I think we 
now can satisfactorily solve this problem and simultaneously gain an insight into the 
origin of the twelve-division of night and day in Egypt, from which eventually our 
twenty-four-division was derived (Sethe, 1920). This progress has been made pos- 
sible by utilizing information contained in a Demotic papyrus of the Roman period, 
purchased by the Carlsberg Fund about twenty years ago for the Egyptological 
Institute of the University of Copenhagen. The late H. O. Lange recognized the 
importance of this text, now called "P. Carlsberg 1", which was then published by 
him and the present writer (Lange-Neugebauer, 1940). In recent years, my 
colleague, Professor R. A. Parker, and I assumed the study of this text in connection 
with our plans for a comprehensive publication of all available astronomical texts from 
Egypt- It was from our discussions that it became clear that the Egyptian texts of 
the Middle and New Kingdom contain all the information required for determining, 
at least qualitatively, the position of the decans and their use for time measurement. 

3. The "decans" make their appearance in drawings and texts on the inner side of 
coffin lids of the tenth Dynasty (around 2100 B.C.). Here we find thirty-six con- 
stellations arranged in thirty-six columns of twelve lines each in a diagonal pattern of 
which the following scheme represents the right upper corner (the columns proceed 

47 



48 Tho Egyptian "Docanft" 

from right to left, as is customary in Egyptian inscriptions). 



The const el hit ions 
"S" are our thirty -six 
"decans". Among them 
figure Sinus; and Orion, 
which, except for the 
Big Dipper, are the 
only two identifiable 
asterisms oi" the Egyp- 
tian sky. 



The use of these "diagonal calendars" was first explained by Pooo ( 1 932), Each 
vertical column serves us a star Work din inn the particular decade the lirsl day of 
which is quoted at the top of the column. For example, the rising of decan S 3 
indicates during the first decade the third hour of the night ; in the second decade, 
the second, eh. When a decan rose in the first hour of the night, it was obviously 
near its acronvchal setting ten days later. (Fig. I may serve as an interesting illustra- 
tion of the decans.) 



day 
21 


day 
11 


day 

1 




s a 


s, 


Si 


hour 




a. 


s, 


hour 






s a 


hour 
3 






ITT 



ft f 



*ltt 



I a 



i n 



rt f 



\\ ] §> £ f$\ * rjth jsj" l r-T If U^T £ ^ , 2 : ? , 



\ \ fj%% & *V4 $*> * ^ "' 



/r, 






"»*■ 






7^ 



/in 






I ±* w 



^nrf ^ 



*"Ti SSSc 



n 



U 






Tf 



1 ■& ^ 'V' -ti- 



AJ 



'£**? 



M< 



Fig. 1. The Details on tlie Coffin of II K l-i. showing bow these arc diagonally 
arranged. {fail 1 *) Mus. 2KI27) 



O. Neugebauer 49 

We shall not discuss the details of these texts, particularly those modifications of 
the scheme which became necessary because of the five epagomenal days at the end 
of the Egyptian year. On the contrary, we shall simplify our discussion by being 
quite unhistorical and replacing ten-day intervals by 10°-segments of the ecliptic. 
The error thus committed has no influence on the interpretation of such crude schemes 
and can be remedied at once if the need should arise. 

Using the rising of stars as indications of the beginning of "hours" means that all 
stars which rise simultaneously — synanatellonta in Greek terminology — are in prin- 
ciple equally serviceable. Thus the diagonal calendars tell us only that the decans 
were stars located at thirty-six positions of the eastern horizon, such that these 
horizons intersect the ecliptic at points that are about 10° apart. 

4. The next bit of information comes from monuments of Seti I and Ramses IV 
(about 1300 and 1170 B.C.), where the decans are represented on the body of the sky 
goddess Nut, and from P. Carlsberg 1, which is an extensive commentary to the often 
very cryptic inscriptions on the monuments. Again we shall not describe details but 
utilize only one fact which became clear only through the ancient commentator: 
all decans are invisible for 70 days between acronychal setting and heliacal rising. 
In other words, all the decans have (at least ideally) the same duration of invisibility 
as their leader, Sirius. This shows clearly the origin of the whole concept of the 
decans. The heliacal rising of Sirius marks, in principle, the beginning of the year, 
and similarly one chose other stars whose rising indicated the beginning of the 
consecutive decades of the Egyptian civil calendar. The gradual removal of each such 
constellation away from heliacal rising was used to mark intervals of the night — we 
shall call them the "decanal hours". And in order to make the whole scheme as 
uniform as possible the selection of the decans from among all simultaneously rising 
stars was made such that they had been seen for the last time 70 days earlier, just as 
Sirius had spent 70 days in the nether world before rising again at the end of the 
last hour of the night. 

5. Before discussing the resulting decanal hours we shall combine what we know 
from the diagonal calendars and from P. Carlsberg 1. The diagonal calendars told us 
that the decans were located on thirty-six horizons which meet the ecliptic at intervals 
of 10°. But a similar condition is imposed on the thirty-six positions of the horizon 
with respect to the ecliptic when the stars set. The two groups must be in the 
relation to each other such that 70 days elapse between corresponding settings and 
risings. The thirty-six intersections between such pairs give the places of the 
thirty-six decanal stars. 

It is easy to carry out this construction graphically. We make, say, a cylinder 
projection of the celestial sphere with the equator as circle of contact (see Fig. 2). 
Let B and A be two positions of the Sun, 70 days apart. Let EH be the position of 
the horizon when a star S rises heliacally, W H the position of the western horizon 
when S sets acronychally. For a star of given brightness, it is known how far the 
Sun in A and B must be distant from the horizon. Thus for given A and B, 70 days 
apart, we can find S at the intersection of the two proper horizons. Moving A and i? 
10 days ahead, we get a new pair of horizons and, at their intersection, the next 
decan. And because Sirius is one of the decans, we are given an initial position AB 
from which to start. Thus all the other decans can be found, at least in principle. 



50 The Egyptian "Decans" 

Of course, this presupposes that we know the brightness of the stars — which is 
obviously not the case except for Sirius. Nevertheless it is clear that less brightness 
must remove the two horizons from A and B and thus bring S closer to the ecliptic. 
And the opposite holds for bright stars. Thus we obtain for the decans a zone, instead 
of a curve, following by and large the ecliptic toward the south, with Sirius located 
at its farthest boundary. 




EH" 
Fig. 2. Graphical determination of the Decans 

6. There would be no point in trying to push the identification of the decans any 
further. What we now know definitely is that both hypotheses which have found 
support among scholars, namely, that the decans are either ecliptical or equatorial 
stars (considering the only certain identifications, Sirius and Orion, as "exceptions" 
to confirm the rule), are equally wrong. It is clear that the 70-day invisibility cannot 
be taken absolutely literally, not to mention the idealizations made by the Egyptians 
in order to maintain the relationship between stars and rotating calendar. All these 
effects soon enough rendered unusable the whole device of the diagonal calendars, 
and it was probably already obsolete in practice when we meet it as the traditional 
time instrument for the use of the dead person on his coffin. Indeed, as Parker has 
recognized, the star tables of the temples of the Ramessides abandon decanal risings 
in favour of transits of quite different constellations. But the decans maintain to the 
end of Egyptian history their role as representatives of the consecutive decades of the 
year and as such they were readily absorbed into the Hellenistic zodiacs. 

7. We must now come back to the "decanal hours". Obviously these "hours", 
determined by the rising of stars on horizons with constant longitudinal difference, 
are neither constant nor even approximately 60 min in length. They vary as the 
oblique ascensions of the corresponding sections of the ecliptic and are about 45 min 
long because during each night eighteen decans rise and set. Thus twelve decanal 
hours seem too short to measure time at night. In so arguing, one forgets, however, 
that each decan has to serve for ten days as indicator of its hour. If we furthermore 
require that these "hours" of the night never should be part of twilight, we get a 
satisfactory covering of the time of total darkness by means of only twelve decans 
during ten consecutive days, especially for the shorter summer nights.* Again for 
the sake of consistency a simple scheme which held true for Sirius near the shortest 
summer nights was extended to all decans and all nights, thus making the number of 



* This can be checked, e.g. by using the tables of oblique ascensions in the Almagest II, 8. 



George Sarton 51 

"hours" twelve for all seasons of the year. And finally the symmetry of night and 
day, of upper and nether world, suggested a similar division for the day. This paral- 
lelism between day and night is still visible in the "seasonal hours" of classical 
antiquity. It is only within theoretical astronomy of the Hellenistic period that the 
Babylonian time- reckoning with its strictly sexagesimal division, combined with the 
Egyptian norm of 2 x 12 hours led to the twenty-four "equinoctial hours" of 
60 min each and of constant length. 



References 

Lange, H. O. and Nettgebauer, 1940 Papyrus Carlsberg No. 1, ein hieratisch- 

demotischer kosmologischer Text, 
Danske Videnskabernes Selskab Hist.- 
filol. Skrifter 1, No. 2. 

Pogo, A 1932 Calendars on coffin lids from Asyut, Isis 

17, pp. 6-24. 

Sethe, K 1920 Die Zeitrechnung der alten Aegypter, 

Nachr. d. Ges. d. Wiss. zu Gottingen, 
Philol.-hist. Kl., p. 28-55; p. 97-141. 



The Astral Religion of Antiquity and the "Thinking Machines" 

of To-day 

George Sarton 

Harvard University, Cambridge, Mass., U.S.A. 



1. The Astral Religion 

The theory of homocentric spheres invented by Eudoxos of Cnidos (IV- 1 B.C.) 1 
made it possible "to save the phaenomena" (aco&tv rh. <poliv6juevoh), that is, to give 
a mathematical explanation of the motions of the "erring" bodies, or planets. 
For example, the spherical lemniscate (iTr-no-nedr]) which Mercury (or Venus) des- 
cribes in heaven could be explained as the cinematical resultant of the motions of 
four homocentric spheres, which are linked together, and each of which rotates 
around its own axis with a different speed. This theory was a magnificent illustration 
of the rationalism of the Greeks, their mathematical genius and their boldness ; they 
were not more shy of introducing incredible postulates in order to give a rational 
account of observations than are the astrophysicists of to-day. 2 

What is stranger still, they were inclined to combine rational and workable theories, 
such as that of the homocentric spheres, with oriental fantasies. Pythagorean 
astronomers were largely responsible for such combinations. Two of their funda- 
mental ideas were very probably of Babylonian or Mazdean origin, to wit, the dualism 
of the universe (sublunar and translunar) and the divinity of the astra, 3 especially the 

1 Symbols like (IV-1 B.C.) or (V-2) mean first half of the fourth century B.C., second half of the fifth century after Christ 

2 Sometime later the theory of homocentric spheres was replaced by other theories making use of epicycles and eccentrics, 
but that did not change the situation. It was simply the replacement of one kinematic explanation by another more convenient! 
The theory of epicycles and, possibly, also that of eccentrics, was originated by Apollonios of Perga (III-2 B c ) 

3 The word astra is used because of its generality. The astra include stars and planets. 



52 The astral religion of antiquity and the "thinking machines" of to-day 

planets. The Pythagoreans inheriting other oriental ideas and, translating them into 
the Greek idiom, added two new conceptions, first the excellence of circular and 
spherical motions, second, the return of men's souls to heaven after death. On 
account of their divinity, celestial motions must be circular; the erratic trajectories 
of the planets across the stars must be explained in terms of cinematic compositions 
of circular movements. In the second place, there is a relationship between astra and 
souls. The immortality of the soul thus became part and parcel of astronomical 
doctrine. 

The endeavour to explain celestial mechanics had led to the formulation of what 
might be called an astronomical theology. The earliest, if not the clearest, formula- 
tion of it was the Epinomis, which as its title suggests, is an appendix to the Laws. 
Was it written or drafted by Plato ? Was it composed or edited by Plato's secretary, 
Philip of Opus (IV- 1 b.c.) ? We have no means of knowing. 

A great historian of ancient religion, the Belgian Franz Cumont, said of the 
Epinomis that it was "the first gospel preached to the Hellenes of the stellar religion 
of Asia". 4 That description is striking, but a little delusive. The astral religion had 
oriental roots, but it was definitely a Hellenic creation, Pythagorean and Platonic. 

I do not claim to understand the Epinomis which is chock full of irrationality and 
is as good an example of pseudo-scientific writing as the Timaios. The gist of it is 
as follows. The aim of wisdom is to contemplate numbers, especially celestial 
numbers. The five regular solids are equated with the five elements (ether being the 
fifth). The most beautiful things are those revealed to our understanding by our own 
souls, or by the cosmic soul and the celestial regularities. The cult of the astra must 
be legalized. Astronomy is not only the climax of scientific knowledge ; it is a rational 
theology. The supreme magistrates are not to be philosophers but rather astronomers, 
that is, theologians. 

The popularity of that astronomical theology and its acceptance for centuries by 
men of great intelligence and wisdom was justified by the following circumstances. 
In the first place, the old mythology had become untenable. The majority of good 
men respected the traditional rites and valued the myths as a kind of national 
and sacred poetry, but they could no longer accept them as true. 5 Hence, there was 
in their hearts a religious vacuum which had to be filled. Secondly, after the loss of 
independence and subjection to Macedonian rule, every Greek was profoundly 
disillusioned. To political servitude were added the tumult of passions and no end 
of economic difficulties. The sublunar world was as chaotic as it could be ; it was the 
home of decay, disorder, corruption, instability, disease, and death. It was insuffer- 
able. There must be another world where order and justice obtain. Thirdly, the 
heavens offer an excellent image of the cosmos. Everything in heaven is regulated. 
The motion of the fixed stars is majestic, the complex periodicities of planetary 
motions are even more awful. This splendor is revealed by mathematical analysis 
which allows astronomers to build up a theoretical reconstruction of the whole 
system, the verisimilitude of which is proved by the comparison of observed positions 
with calculated ones. The concordance is imperfect, but such as it is, it could not be 
accidental; it is imperfect but indefinitely perfectible. 



4 F. Ctjmont; Astrology and Religion among the Greeks and Romans (Xew York, Putnam, 1912), p. 51. 

* Lucian of Samosata made fun of the myths, but he was very exceptional, a child of the Euphrates, and came much later 
(fl. c. 120-200). He has been called the Voltaire of antiquity. The average Greek gentleman («:a/lo/<aya0o?) attended the 
ceremonies and mysteries and kept silent, but he could not believe the stories. 



George Sarton 53 

Hence, the final conclusions were readily assented to. The stars and planets are 
the homes of the Gods and the final retreats of human souls. 

We may still add that the Epinomis was an intrinsic part of the Platonic canon 
and partook in its authoritativeness and its glory. This was the result of a series of 
misunderstandings and unconscious prevarications. Non-mathematicians, that is 
the majority of the people, accepted Plato's conclusions as corollaries of his mathema- 
tical work 6 ; the mathematicians accepted it as a kind of metaphysics beyond their 
own ken. Plato's reputation for wisdom was so great that everybody assumed he 
had established the validity of his statements, however fanciful they might be. 

Hence, the astral religion was countenanced by almost every thinker of the 
Hellenistic and Roman ages. It was incorporated into the fabric of Stoicism and 
explained by Poseidonios of Apameia (1-1 B.C.); Poseidonios' disciple Cicero 
(1-1 B.C.) re-explained it to Latin readers. Cicero wrote good accounts of it in the 
Somnium Scipionis and in the Be natura deorum. The astral religion was the religion 
of every educated man who could not swallow any longer the traditional mythology 
and had not been converted to another faith. In particular, we may say that it 
formed the natural intermediary between the doctrines of Paganism and those of 
Christianity. 

The Somnium Scipionis preserved by Macrobius (V-l) had a deep influence upon 
Christian thought. 7 It did not transmit the whole of astral theology to the western 
world, but it offered a kind of proof of the immortality of the soul which was as highly 
acceptable to Christian theologians as it would have been to Jewish or Muslim ones. 
That is another story, however, which does not concern us at present. 

For some six centuries (three before and three after Christ) the astral religion was 
the religion of the elite of the Pagan world. It was accepted by the wisest men, 
such as Marcus Aurelius. One may wonder why Jewish or Christian doctrines did 
not reach the Pagan world earlier. The answer is simply that the Pagans, the 
Christians, and the Jews hardly mixed even in the great cities where they lived 
together. For example, Galen was a very learned man, the kind of man who knew 
everything ; yet, his knowledge of Judaism and Christianity was so rudimentary 
that he was hardly able to distinguish between them. 8 This is the more astonishing, 
because he must have come across many Christian and Jews in Pergamon, Alexandria, 
and Rome, and he actually used the commentary on Epidemics by Rufus of Samaria, 
who flourished in Rome at the same time as himself. Similar remarks might be made 
about Marcus Aurelius. Jews and Christians did not mix with Pagans any more 
than they could help and the revulsion was mutual. They were minority groups 
binder suspicion of disloyalty; the intolerant considered it their duty to denounce 
and persecute them, while the more tolerant preferred to steer clear of them. The 
unfortunate circumstances of our own time make it easier for us to understand 
that situation. 

It is not difficult for us to appreciate and even to consider with sympathy the attitude 
of mind of those enlightened Pagans who found solace in celestial mechanics. It 
did not occur to them (and it would be unfair to blame them) that the trajectory of 

n r ' I^f?«« i P ple ' , th f cos mological meaning of the five regular solids, which is reaffirmed in Epinomis, was accepted as a piece 
SlS^^%^r e i^ffit f Ma ffi a d r°g V u e S g ° f the ^ S0MS W " h the ^ eWentS wa^fantastic^ 

w^^t^S.^ Z:^^^S^^^X£, 1 !^V'^^T y wel1 annotated Dy 

KICHARD Walzer: Galen on Jews and Christians (102 pp., Oxford University Press, London, 1949). 



54 The astral religion of antiquity and the "thinking machines" of to-day 

the Earth observed by a Venerian astronomer would have been as pure and sublime 
as the trajectory of Venus as seen by themselves. The main trouble with the sub- 
lunar world is that we are too close to it for abstractions and illusions. We under- 
stand their feelings of sacredness which were comparable to Kant's, writing in 1788, 
"Two things fill one's conscience with ever increasing wonder and awe, the stars in 
heaven and the moral law in oneself". Yet, Kant was deeply moral and rational, 
while they were demoralized and overwhelming miseries had driven them into 
irrationality. 

There is one thing, however, which puzzles me very much. The astronomical 
theologians admired the celestial order so much that they identified it with intelli- 
gence, and, on the other hand, they tended to identify disorder and freedom with 
stupidity (Epinomis 982-3). The planets reveal divine intelligence by the eternal 
accuracy of their motion. Now we might admit with Plato that the planetary 
motions reveal God, but not that the planets themselves are gods. Think of the 
popular argument of the clock. Its mechanism and regular motions reveal the 
existence of a clockmaker. Nobody ever said that the clockmaker was in the clock, 
or that the clock was itself the clockmaker. Yet, according to the new astrologic 
religion, the planets did not simply reveal God, they were themselves gods, each planet 
regulating its own motion with divine intelligence and repeating it eternally to 
evidence its own wisdom. Does that make sense 1 Yet the argument was accepted 
by the New Academy and by the Stoics and we find it very clearly stated by Cicero 
(De natura deorum II, 16). The confusion of thought was probably caused by a wrong 
generalization; the soul or intelligence of an animal is within itself; we may say 
that the animal has intelligence or that he is an intelligent being; his intelligence, 
however, is revealed not by the regularity and precision of his motions, but rather 
by their unexpectedness. 

How could these wise people confuse automatism with intelligence and freedom 
with foolishness? Is it not tragic to witness Greek philosophy ending in such a 
shameful impasse ? 

A similar confusion occurring under our own eyes may help us to judge that 
paradoxical situation with indulgence, but before speaking of that it is worth while 
to end our account of the astral religion with a brief discussion of the week. 

The best proof of the popularity of the astral religion during the decline and fall of 
Paganism is given by the establishment of the week. 9 The number of days in the 
week is seven, because of the recognition of seven planets which were generally listed 
in the following order : Saturn, Jupiter, Mars, Sun, Venus, Mercury, Moon. This is 
proved by the fact that the names of the days are derived from planetary names in 
many languages, and because the order of the week days is easily deductible from the 
normal order just given. 10 What is astounding and could not have been foreseen is 



9 For the history of the week, see Francis Henry Colson: The Week (133 pp., Cambridge University Press, 1926) and 
Solomon Gandz: "The Origin of the Planetary Week in Hebrew Literature" (Proc. American Academy of Jewish Research, 
18, 213-54, 1949). 

10 It is the order of what they believed to be decreasing distances from us ; the order was correct except, of course, for the Sun. 
The week order is deduced from the normal order by taking the first and dropping two, then the second and dropping two, etc. 
One finally gets the series, Saturn, Sun, Moon, Mars, Mercury, Jupiter, Venus. The Jewish week was essentially different from 
the astrological week, except that both had seven days. The first day of the Jews was Sunday and the last Sabbath. The first 
day of the astrological week was Saturday, an unlucky day, while the last Jewish day, Sabbath was a day of blessing. 

The days of the week were called after the Planets or Gods which dominated their first hour ; the following hours were dominated 
by the following planets in the normal order. Thus, the first Pagan day, Saturday (Saturni dies) was so called, because its first 
hour was dominated by Saturn; the second hour was dominated by Jupiter; the third, tenth, seventeenth, and twenty -fourth 
hours were dominated by Mars. The first hour of the second day (Solis dies) was thus dominated by the Sun, etc. If one con- 
tinues the calculation, one obtains the astrological order of the days: Saturday, Sunday, Monday, Tuesday, Wednesday, 
Thursday, and Friday (the astrological origin of the days' names is more obvious in many other languages, e.g. in Italian). 



George Sarton 55 

that the astrological week was diffused unofficially throughout the Roman empire 
not long before Christ and about the beginning of our era. 

The six days of creation followed by the sabbath equalled in number the seven 
planets. That was a pure coincidence, but it favoured the success of the astrological 
week. At any rate, the hebdomadary period, if not always the names of the days, 
spread gradually all over the world. 

Next to the silent adoption of the decimal basis of numbers, it is the most remark- 
able example of unconscious convergence and unanimity in the whole history of 
mankind. 

The establishment of both periods was certainly facilitated by the fact that it was 
not deliberate. If one had tried to establish it by means of international congresses 
(assuming that that would have been possible or even conceivable in those early 
days), the chances of failure would have been considerable, because the two periods 
unconsciously agreed upon, ten and seven, are good enough but not absolutely so. 
It is clear that a week of six, eight, or ten days would have been equally acceptable, 
and the number bases eight or twelve have as much to recommend themselves as 
the basis ten. On the other hand, it is just as well that the convergences were not 
carried too far; logicians might have been tempted to promote a decadic week 
instead of the hebdomadary one in order to agree with the decimal system, and we 
would have lost three holidays out of every ten. 

In India the hebdomadary succession of days was accepted together with much 
else in Greek astrology. Witness the Surya-siddhanta which dates probably from the 
fourth or fifth century. 11 In early China, however, where Greek astrology did not 
penetrate, there seems to have been a decimal week, and much later, under the 
T'ang (621-907) officials were allowed one holiday in every decade of days (hsiin 
hsiu). 12 The seven-day week was introduced into China about the eighth century by 
Manichaeans, by Nestorians or by Hindus 13 ; it did not find as wide an acceptance as 
in the western countries. 

Bearing in mind the Chinese anticipation, the decimal week introduced by the 
French Revolution was not a complete novelty. It did not last very long, only 
fifteen years (1792-1807); its unpopularity was probably aggravated, if not caused, 
by its very length. The period seven is more in agreement with physiological needs ; 
a stretch of six working days is ample, 14 we would hate to have it increased 50 per 
cent for the sake of decimal consistency. 

The cultural convergence which caused the establishment of the week and the 
decimal basis is even more mysterious than the linguistic convergence making possible 
the birth of a language. In the linguistic cases only a relatively small group localized 
in a relatively small region was concerned while the diffusion of the week and of the 
decimal basis implied unconscious collaboration on an international, interracial, 
interreligious, and polyglottic scale. On the other hand, it must be admitted that any 
language, even the least developed, is a very complex fabric implying agreement on a 
very large number of words and rules. 

11 Sarton: Introduction (1, 386-8) 



Goodrich, of New York, and for Sanskrit information to my friend, Paul Emile Dumont, of Baltimore 

1 or Chinese decimalism in cartography, see my Introduction (2, 225). 

13 Sarton: Introduction (1, 504, 513-14). 

I4 + F< 2 P, e °P le who , re 5 1] y work! A missionary told the story of primitive men converted to Christianity who rested with 
great affectation on Sundays in order to suggest that they did actually work on the other days. Raoul Allibr: La Psychologie 
de la Conversion chez les Peuples Non-civUisis (2 vols., Paris, Payot, 1925; Isis 30, 306-10) 



56 The astral religion of antiquity and the "thinking machines" of to-day 

2. "Thinking Machines" 
The history of science, and even the history of mankind, could be written in terms of 
tools and artifacts. The stone axes of our Palaeolithic ancestors were tools, and so 
were the wheels, pulleys, microscopes and telescopes, thermometers, dynamos of later 
times. The arms and armour of all nations from the club to the atomic bomb are 
tools of another kind. 

One might claim that the history of science is a history of ideas, and that claim is 
correct ; yet every scientific idea (with the possible exception of very abstract or 
mathematical concepts) led to the creation of a new instrument or of a better one. 
If books and prints are included among the instruments, there is no exception of any 
kind. 

Every tool is an extension of a human organ or the amplification of a function, 
material or spiritual. Many of them amplify our mechanical power, some amplify 
it exceedingly; they may multiply it a million times or more. Others increase the 
acuity of our vision ; telescopes and microscopes increase it so much that they reveal 
new worlds to us, infinitely large or infinitely small. If photographic plates are 
properly attached to those instruments, they serve to amplify immeasurably our 
memory. Of course, every book or print helps to do the same ; a good library is the 
memory of mankind. 

No tool is worth anything unless there be somebody able to use it, and when a tool 
or, let us say, a machine is sufficiently complex, it must be guided or restrained by a 
workman. Ingenious people, however, found means of contriving the machine to 
help or guide itself. Humphrey Potter, the lazy boy who got tired of opening and 
closing valves in an early steam engine made the engine self-acting by causing the 
beam to open and close them with suitable cords and catches. That was as early as 
1713. Not only was the work done, but no human being could have done it as faith- 
fully. Another early example was Watt's governor (1784) controlling automatically 
the speed of an engine, and this type of governor (pendulum governor) was followed 
by many others. An admirable kind of servo-mechanism was invented in 1799 by 
Joseph Marie Jacquard (1752-1834) of Lyon. Jacquard looms made it possible 
to weave a fabric with the most complicated pattern as easily as a plain one. His 
invention opened endless possibilities, for his cards and looms could be modified 
indefinitely. 

The mechanical governors represented a type of contrivance which was extended 
later to many other fields ; the purpose of all of them was to establish a kind of 
homeostatic control, for example, thermostats and voltage stabilizers. 

To return to the eighteenth century, many machines had been invented before the 
end of it ; and a variety of servo-mechanisms had been introduced. There were also 
calculating machines and slide rules of various kinds. In our own days computing 
mechanisms and linkages have been carried to such a high stage of perfection that 
innocents are beginning to speak of them as "thinking machines". The extraordinary 
speed and accuracy of those machines and their fast increasing "memory" suggest 
comparisons with human brains. 

The reference to them as "thinking machines" (a reference which was never made 
with regard to Jacquard looms) was caused partly by the fact that ignorant people, 
mistaking computation with mathematics, assumed it to be a highly intellectual 
function, 15 partly by the lightning speed and precision of those machines. From that 

15 For example, lightning calculators or outstanding chess-players are often believed to be men of great intellectual genius. 



George Sarton 57 

particular point of view, electronic computers are not human but superhuman, for 
they approach a perfection hopelessly out of man's reach. (But are the strength of a 
motor, the visual power of a telescope or microscope not superhuman in the same 
way?) 

No machine is able to do anything by itself without human initiative. In that 
respect, an electronic computer is on the same level as any other tool. The confusion 
exists, however, and it is pernicious. Some enthusiasts are dreaming of machines 
which would play chess, translate from one language into another, or compile 
bibliographies. To return to reality, it is clear enough that machines could be invented 
to accomplish the most servile parts of any undertaking, machines which could do 
everything, except the essential. 

Outside of the mathematical aspect (which is always full of mystery and glamour 
for the non-mathematician), the feature which impresses the popular mind most 
deeply is the contrivance of a mechanical memory, comparable to human memory, 
though with an infinitely greater capacity and precision. Such a memory may be 
realized by (1) magnetic drums (any of 81,920 digits can be stored or recalled in an 
average of J5 of a second), (2) cathode ray tubes (any of 10,240 digits can be stored 
or recalled in 10" 5 sec, (3) magnetic tapes (any of 2,000,000 digits can be stored on 
one tape and recalled from it at the rate of 12,500 a second). 16 

Dr. Vannevar Bush was so impressed by various mnemonic contrivances that he 
coined the ingenious phrase "memex instead of index". 17 Memory is certainly one of 
the fundamental components of intelligence, but it has gradually lost much of its 
importance, because mankind has created artificial treasures of information which 
are so much superior to natural memories that the latter may be disregarded. Men 
endowed with phenomenal memories such as existed in the past and still exist in 
countries like India, 18 are no longer looked up to as if they were men of miraculous 
wisdom, but looked down upon as idiots. Memory will always be indispensable as a 
general guide, but the emphasis has changed from literal memory to a broader and 
vaguer kind. A man of science relies on his memory but does not trust it overmuch ; 
he does not try to remember special data, but rather to know the books wherein those 
data are exactly recorded. 

There are in any kind of material or intellectual work an endless number of 
repetitive processes which could (and will) be done by machines incomparably better 
than by men. Not only are many operations susceptible of being carried out by a 
machine, but servo-mechanisms may enable us to accomplish series of operations of 
indefinite length and complexity, and to control groups of simultaneous operations 
or of simultaneous series of operations. There is no limit to such a development. 
As soon as the greatest and most efficient machine has been invented, 19 it is relatively 
easy to invent one that is more complex, more versatile, and more efficient. 

The ancients worshipped the planets because of the awful regularity of their 
motions. It is well that our contemporaries are satisfied to boast of their "thinking 
machines" (which are far more regular than any planet could ever be) and have not 
yet ventured any apotheosis. 



" The figures quoted represent the realizations of to-day (May, 1953), as given by International Business Machines, New 
York. Present possibilities can be and will be indefinitely increased. 

17 V. Bush: Endless Horizons (p. 31, 1946). 

18 Examples and references given in my History of Science (vol. 1, 132, 1952). 

V 'l In Xfr n ^5"i. not " built "- A better machine will be invented before the preceding one is actually built. As Le Corbeillbr 
puts it, Well before the machine is Anally built, the designers have in mind an entirely different principle and twenty improve- 
ments of detail which, when incorporated into the next type, will, they tell you, make the present one 'look like peanuts' " 



58 The astral religion of antiquity and the "thinking machines" of to-day 

3. Conclusions 

Every "thinking machine" needs a human brain and human hands behind it or it 
cannot think at all. Many problems may be solved with lightning speed and unerring 
precision, but who will formulate them ? Machines can answer questions exceedingly 
well, but who will ask the questions ? If a long series of operations is determined by a 
sequence of holes punched in a ribbon, there must still be somebody to punch the 
holes. Some machines are exceedingly versatile, but versatility should not be 
confused with freedom. 20 

As far as performance goes, the wonder of our time is less the giant computer than 
the means of reproducing music, synchronizing it with pictures and transmitting the 
combination to any distance. Curiously enough, the tools which make television 
possible are never assumed to be "thinking" ; that supreme quality is assumed to be 
restricted to the computing machines, because a great number of people mistake 
computation for thought. Let us repeat that real thought is free, and that com- 
putation is the opposite of free; it is servile. 

In both cases — the ancients adoring the planets and the children of our time 
standing in awe before the "thinking machines" — there is an identical confusion 
between regularity and precision, on the one hand, and intelligence on the other. 21 

Let us reverse the argument and say : everything that is perfectly regular betrays 
a lack of intelligence. The essential difference between the average man and the 
perfect robot is that the former can and will think, however little, while the latter 
cannot think at all. 

The paradox which misleads people lies in this. Human thought is free, spontaneous, 
unpredictable; yet we expect it to develop along logical patterns. It must be logical 
and consistent, free of arbitrariness, but not too much so. The same is true of every- 
thing beautiful; beauty implies complex symmetries, not perfect ones. 

Most of the paradoxes of life, art, and virtue hinge on that one. Think of the main 
treasure house of any culture, and if one may use the metaphor, its main power house, 
language. Each language is the combination of innumerable rules and words; 
the words hang together in capricious sequences ; the rules are dovetailed in many 
ways, but the meanings of many words are erratic and the rules are full of exceptions. 
The elaboration of thought in a single head is mysterious enough, but how were the 
languages developed, each of them simultaneously in thousands of heads, without 
deliberation and almost without consciousness ? Now, that miracle, the creation of a 
language, implying the web and woof of capricious rules and the combination of 
logical with illogical patterns, has not happened once but thousands of times. 

Or look at it from the point of view of a student. If it be our maternal language, 
we learn its rudiments intuitively, without analysis. If our mother spoke it purely, 
we learn to speak it with equal purity. The Greeks wrote their masterpieces before 
any grammar had been composed; indeed, the early grammars were built on the 
basis of those masterpieces, not vice versa. If we have to learn a foreign language 
later in life, we proceed more methodically. We try to build up a sufficient vocabulary 



20 As does Lb Corbeiller. The automatic telephone is very versatile but has no freedom ; all the freedom is in the head 
of the person who dials a given number. That number is a definite order which the telephone has no choice but to obey ; that 
is why it is called automatic. 

21 One might claim that the ancients were aware of the fact that planetary motions are not perfectly regular. The planets of 
heaven do not move as regularly as those of an orrery. The ancient astronomers were aware of irregularities when computed and 
observed places did not coincide ; their reaction, however, was to conclude, not that the planets were irregular but that their 
cinematic explanations were imperfect. 



George Sarton 59 

and to master the rules of morphology, accidence, and syntax. Yet, as long as we 
have to think of the dictionary and the grammar of a language, we can hardly use it 
in a fluent way. We are beginning to know a language only when all the words and 
rules have been driven into our unconsciousness; we know it only when we have 
reached a high degree of automatism in our use of it. Automatism in this case is a 
measure of progress, but not for very long, for no one can use a language supremely 
well unless he does so with great deliberation. 

Consider the problems of administration. Every administrator dreams of regular- 
izing his business as much as possible, of automatizing it, but he could never succeed 
unless the business entrusted to his care were almost dead or he were himself excep- 
tionally stupid. Such insensitive and dictatorial administrators exist, but they are as 
amiable as Procrustes. The paradox lies in the fact that administration must be 
very orderly yet human, which means disorderly; it must be impartial yet sensitive, 
and this implies love and partiality. 

Consider art. Gilbert Murray hit the nail on the head when he wrote: "The 
unwillingness to make imaginative effort is the prime cause of almost all decay in art. 
It is the caterer, the man whose business it is to provide enjoyment with the very 
minimum of effort, who is in matters of art the real assassin". 22 Now, any imaginative 
effort or creative effort is an escape from automatism, a rebellion against it, a 
defence of personality and of freedom. Dictators in art as well as in politics arise 
only because too many men are spiritless arid timorous and prefer to eschew the 
difficulties of choice and the anxieties of decision; such men, incipient robots, betray 
humanity; they are more like sheep and pigs than soulful beings. 

Consider education. It is to some extent a school of automatism. Educators try 
to inculcate good habits. We all need good habits. The business of life is so complex 
that we must learn to do many things instinctively and to reserve our thinking 
power for things of greater worth whileness. Our physiological functions are largely 
automatic ; if they were not so, we could not exist. We must try to automatize other 
functions, material and spiritual ; this will enable our mind to soar off from a higher 
level. Habits are activities which have become gradually unconscious; they are 
necessary but very insufficient. A man whose life is too regular and who is dominated 
by habits is not admirable at all but contemptible. 

Success in life, in art, in virtue always implies a victory of freedom over automatism, 
of new creation over dead-like repetition. 

It is well that we have with us enough engineers who are obsessed with the vision 
of new gadgets, for gadgets may be very useful, but the real business of life is 
independent of them. 

There will always be a need for "thinking machines", better ones and more of them, 
but the greatest need is for thinking men, not robots nor even "yes-men" but 
independent men, honest and free. 



Short Bibliography 

1. Astral Religion 

Cumont, Franz (1868-1947): Les Religions Orientales dans le Paganisme Roniain (4th edition, 355 pp 

16 pi., 13 figs., Paris, Paul Geuthner, 1929; Isis 15, 271). 
Cumont, Franz (1868-1947): Lux Perpetua (562 pp., Paris, Geuthner, 1949; Isis 41, 371). 
Epinomis : Greek text and English translation by W. R. M. Lamb in the Plato edition of the Loeb series 

(vol. 8, 426-87, 1927). 

" Religio Grammatici (London, Allen, 1918), p. 34. 



60 Ptolemy's precession 

Rougier, Louis : L'origine Astronomique de la Croyance Pytfmgoricienne en V Immortalite Celeste des Ames 

(152 pp., Recherches d'archeologie, de philologie et d'histoire, 6; Institut francais d'archeologie 

orientale, Le Caire, 1933; Isis 26, 491). 
Sarton, George: A History of Science: Ancient Science through the Golden Age of Greece (672 pp., 

103 figs., Harvard University Press, Cambridge, Mass., 1952), pp. 447-54. 
Sarton, George: Introduction to the History of Science (3 vols, in 5, Baltimore, Williams & Wilkins, 

1927-48). 

2. "Thinking Machines" 

Berkeley, Edmund C. : Giant Brains, or Machines that Think (286 pp., New York, Wiley, 1950). 

Bush, Vannevar : Endless Horizons (192 pp., Washington, D.C., Public Affairs Press, 1946 ; Isis 37, 250). 

Diebold, John: Automatism. The advent of the automatic factory (182 pp., New York, Van Nostrand, 
1952). 

Le Corbeiller, Philippe: "Large-scale Digital Calculating Machinery" (American Journal of Physics, 
16, 345-7, 1948). 

Lemoine, Jean Gabriel: "La 'Machine a Penser' de Raymond Lull et l'Astrologie Arabe" (Bulletin 
de la Societe de philosophic de Bordeaux, 5e annee, No. 25, pp. 55—64, Aout 1950). 23 

Proceedings of two symposia on large-scale digital calculating machinery (vols. 16 and 26 of the Annals o 
the Computation Laboratory of Harvard University, Harvard University Press, 1948, 1951). 

Svoboda, Antonin: Computing Mechanisms and Linkages (No. 27 of Massachusetts Institute of Tech- 
nology Radiation Laboratory series, 372 pp., New York, McGraw-Hill, 1948). 

Willers, Friedrich Adolf : Mathematische Maschinen und Instrumente (340 pp., 258 figs., Berlin, 
Akademie-Verlag, 1951). 



23 This item is mentioned, because the curious use of the phrase "thinking machine" in its title. There is a difference, 
however, which it is worth while to explain. Every branch of logic or mathematics might be called a "thinking machine", 
because it helps us to think somewhat automatically. For example, given a system of equations, one proceeds to solve it 
according to the rules of algebra or analysis and without bothering about their physical meaning. Whether these equations be 
solved with paper and pen or with more complicated tools is immaterial. The main point is that their solution may lead even- 
tually to the solution of a physical problem and to a discovery. Ramon Lull (XIII-2) was trying to create such a machine, 
logical rather than algebraical (Introduction, 2, 901-4). 



Ptolemy's Precession 

A. Pannekobk 

Formerly Sterrekundig Instituut, Universiteit van Amsterdam, Netherlands* 

Accoeding to Ptolemy's great astronomical work, the Mathematical Composition, 
the movement of the equinoxes relative to the stars was discovered by Hipparchus. 
Probably it had been suggested by his result that the length of the year, as a cycle of 
seasons determined by the equinoxes, was shorter than the return to the same 
phenomena of the stars. This was explained by the assumption that the sphere of the 
fixed stars had a movement along the ecliptic in the same direction as the movement 
of the planets. 

To confirm this explanation Hipparchus made use of observations of the stars 
themselves. Ptolemy says : "In his treatise 'On the variation of the solstices and 
equinoxes', Hipparchus by comparing lunar eclipses observed in his time with others 
observed formerly by Timocharis, arrives at the result that Spica in his time stood at 
6° distance, and in Timocharis' time stood at 8° distance before the autumnal 
equinox. . . . And also for the other stars which he compared he shows that they 
have proceeded by the same amount in the direction of the zodiacal signs" (Book VII, 
Chapter 2). The advantage of using lunar eclipses is that the direct comparison of 
the star with the moon gives the place relative to the Sun which, through the tables 
or by direct comparison, can be easily reduced to the equinox. The years between 
which the change of 2° took place are not indicated in the quotation. Total lunar 

* Now at Wageningen, Netherlands. 



A. Pannekoek 61 

eclipses near Spica happened in Timocharis' time in 283 b.c. on 7 March, and in 
Hipparchus' time in 135 b.c. on 21 March. In Book III, Chapter 1, dealing with the 
length of the year, the latter is mentioned with the addition that a distance of 5|° 
between Spica and the equinox resulted therefrom ; in addition, another eclipse in 
146 b.c. on 21 April is mentioned with a resulting distance of 6|°. If these are the 
source of the rather cursory statement of the quotation, then a change of 2° in an 
interval of, say, 283 — 140 = 143 years, means a precession of 50" per year. 

Ptolemy does not give this computation. Immediately after the sentence quoted 
above he continues by describing his own measurements which, he says, are in 
accordance with Hipparchus' result. With his "astrolabon" (armilla) he determined 
the longitude difference between the Sun and a star, first measuring the distance 
between the setting Sun and the Moon, and then when darkness had fallen, between 
the Moon and the star. "As an instance" he gives the data of an observation of 
Regulus in a.d. 139 ; he finds its longitude to be 122° 30' which, compared with the 
longitude of 119° 50' found by Hipparchus, affords a displacement of 2° 40' in about 
265 years or 1° in 100 years. "This was also assumed with reserve by Hipparchus, 
who in his treatise 'On the length of the year' says '. . . when therefore the solstices 
and equinoxes in one year are retrograding at least 1/100 degree against the direction 
of the signs, they must have receded in 300 years by at least 3°'." Ptolemy continues 
that "After having observed in the same way also Spica and the brightest stars near 
to the ecliptic . , ."he found from them the same amount. Thus with the omission 
of the restriction "at least" he assumes 1° in 100 years (36" per year) to be Hipparchus' 
value of the precession. 

It has often been assumed that Ptolemy simply copied and took over this value 
from Hipparchus and doctored his own observations to bring them into accordance 
with it. It has been made clear, however, by Dreyer* that the wrong amount of 
2° 40' can quite well have been the genuine result of observations affected by an 
accumulation of systematic errors (refraction, solar tables, instrumental errors) 
which, moreover, were working on different stars in a similar way. 

Ptolemy's Chapter 3 of Book VII is devoted to a more detailed test of Hipparchus' 
assumption that it was a movement of all the stars about the pole of the ecliptic. 
Good results could be expected because he had observations from three epochs; 
those of the early Alexandrian astronomers Timocharis and Aristyllus, those of 
Hipparchus, and finally his own, thus extending over a far longer interval. The 
distances in latitude from the ecliptic he found to be nearly equal to what they were 
at the time of Hipparchus. The distance to the equator, however, measured along a 
circle through the celestial poles, was different for the three times of observation. 
For the stars nearer the vernal equinox the new positions were farther to the north 
than the older ones, and for those nearer the autumnal equinox, farther to the south. 
To show this clearly he gives, for eighteen stars, the distance to the equator (our 
declination) as measured by the observers at the three epochs. 

In the ensuing discussion he says that the precession of 2° 40' in 265 years between 
Hipparchus and himself will appear clearly in the differences (of declination) for 
stars in the vicinity of the equinoxes. He chooses six stars for which he gives the 
observed difference and for each of them states, without giving details, that the same 
difference of declination is found for two points of the ecliptic situated about the 
star's longitude at a mutual distance of 2° 40'. The latter quantity can be easily 

* J. L. E. Dreyer, M.N., 77, 536 (1917). 



62 



Ptolemy's precession 



read from the table in Book I, Chapter 15, giving the declination for every degree of 
longitude of the ecliptic. We find for : 



Observed difference . 
Computed difference . 



j? Tauri 



+ 65' 
+ 56' 



Capella 



+ 46' 
+ 39' 



Bellatrix 



+ 42' 

+ 42' 



Spica 



+ 66' 
+ 64' 



r\ Ursae 
Ma j oris 



+ 65' 
+ 56' 



+ 70' 
+ 64' 



The concordance must be deemed satisfactory, especially if we consider that the 
computed values hold for stars in the ecliptic and should be increased for stars of 
higher declination. 

The catalogue of declinations of eighteen stars given by Ptolemy deserves a 
closer examination, because it can be put to a wider use than that of confirming 
qualitatively the course of the precession and to test Ptolemy's supposition for some 
few cases. It can provide us with a more quantitative knowledge of the precession 
as well as of the character and accuracy of Greek observational astronomy. It 
represents wellnigh the only collection of first-hand data left in the literature which 
can give information on the measuring work of the first Alexandrian astronomers as 
well as of Hipparchus. Whereas we know almost nothing about instruments used in 
those ancient times, so that it could even remain doubtful whether they had done 
anything more than observe the phenomena of eclipses and equinoxes, we have here 
the results of measurements with graduated circles and expressed as "distances in 
latitude to the equator on a great circle through its poles". 

In Table 1 after the names of the stars in the first column the six stars selected 
by Ptolemy are marked by an asterisk ; the change of declination by the precession is 
given in the second and third column, first between the early Alexandrians and 
Hipparchus, and secondly between Hipparchus and Ptolemy. They are positive at 

Table, 1. Comparison of declination differences 





Observed declina- 








Computed declina- 




( n ..iv< ■» 




tion difference 








tion difference 


■terrors ^uum./ 


Star 






(for 100 B.C.) 


cos a 


























H.-Al. 


Pt.-H. 








H.-Al. 


Pt.-H. 


H.-Al. 


Pt.-H. 


1. Altair 


0' 


+ 2' 


271° 


58' 


+ 0-034 


+ 2' 


+ 3' 


- 2' 


- 1' 


2. rj Tauri* 






+ 40 


+ 65 


27 


13 


+ 0-889 


+ 48 


+ 79 


- 8 


- 14 


3. Aldebaran 






+ 60 


+ 75 


39 


53 


+ 0-767 


+ 41 


+ 69 


+ 19 


+ 6 


4. Capella* 






+ 24 


+ 46 


42 


38 


+ 0-736 


+ 40 


+ 66 


- 16 


- 20 


5. Bellatrix* 






+ 36 


+ 42 


53 


41 


+ 0-592 


+ 32 


+ 53 


+ 4 


- 11 


6. Betelgeuse 






+ 30 


+ 55 


60 


47 


+ 0-488 


+ 26 


+ 44 


+ 4 


+ 11 


7. Sirius 






+ 20 


+ 15 


78 


9 


+ 0-205 


+ 11 


+ 18 


+ 9 


- 3 


8. Castor 






+ io 


+ 14 


79 


31 


+ 0-182 


+ io 


+ 16 





- 2 


9. Pollux 









+ io 


83 


31 


+ 0113 


+ 6 


+ io 


- 6 





10. Regulus 






- 40 


- 50 


123 


7 


- 0-546 


- 29 


- 49 


+ 11 


+ 1 


11. Spica* 






- 48 


- 66 


174 


21 


- 0-995 


- 54 


- 89 


- 6 


- 23 


12. rj Ursae* 






- 45 


- 65 


184 


42 


- 0-997 


- 54 


- 89 


- 9 


- 24 


13. t Ursae 






- 45 


- 90 


177 


17 


- 0-999 


- 54 


- 89 


- 9 


+ 1 


14. e Ursae 






- 54 


- 81 


166 


32 


- 0-973 


- 53 


- 87 


+ 1 


- 6 


15. Arcturus* 






- 30 


- 70 


189 


54 


- 0-985 


- 53 


- 88 


- 23 


- 18 


16. a Librae 






- 36 


- 94 


194 


53 


- 0-966 


- 52 


- 86 


- 16 


+ 8 


17. /? Librae 






- 48 


- 84 


201 


56 


- 0-928 


- 50 


- 83 


- 2 


+ 1 


18. Antares 






- 40 


- 75 


216 


40 


- 0-802 


- 43 


- 72 


- 3 


+ 3 



A. Pannekoek 



63 



the vernal, negative at the autumnal side. To the modern astronomer it offers a set of 
data from which he can directly derive the constant of precession, since A<5 = number 
of years X n cos oc, where n = precession X sin e (obliquity). There is an uncertainty 
in the number of years, which is hardly relevant; in what follows we took 265 
years (between + 137 and — 128) from Ptolemy, and took 287 B.C. as a mean 
earliest date. In Columns 4 and 5 the right ascension a for 100 B.C., and cos a, 
are given. Taking the sum total of all Ad and all cos a, separately for the positive 
and the negative side, and then combined, we have : 





EA<5 


S cos a 


160« 
H.-Al. 


265n 
Pt.-H. 


n 




H.-Al. 


Pt.-H. 


H.-Al. 


Pt.-H. 


1-9 

10-18 

1-18 


+ 220' 

- 386' 

606' 


+ 324' 

- 675' 

999' 


+ 4-006 

- 8-191 

12-197 


55' 
47' 
49 '■! 


81' 
82' 
81 '9 


0'34 
0'29 
0'311 


0'31 
0'31 
0'309 



The results n = 18-7 for the first, n = 18-5 for the second interval, correspond to 
precessional constants of 46"4 and 46''0. 

This is the value of the precessional constant derived from the observational data 
presented in Ptolemy's book. Ptolemy himself could not make such a derivation, 
since exact formulae giving the dependence of A<5 on the position were lacking, as 
well as the right ascension of the stars. There is no sense in making a solution by 
the method of least squares to see what errors remain, for we are able to derive the 
real errors of the observed changes in declination, since we know the true precessional 
constant. With the true n = 20-24 = 0-337, 160n = 54^0, 265n = 89^4, we find 
the values in column 6 and 7 ; the next two columns give the errors in the numerical 
values of the observed differences of declination. The resulting square mean is 
Vl09 for the first interval of time, a/135 for the second, so Vl22 = 11' is found as 
the mean error of the difference between two ancient observers, and 8' as the mean 
error of a declination measured by one of them. 

The distribution of the values and the signs shows that we have here a normal 
set of observational data, apparently unbiased and undoctored, and not marred by 
systematic errors from faulty tables of the Sun or Moon. We can study this apart 
from the precession problem, by comparing the observed declinations themselves with 
the real values deduced from modern data. For the years 289 B.C., 129 B.C., and 
a.d. 137, assumed to represent the mean dates, the values were interpolated from 
Neugebauer's tables (1912). The comparison is given in Table 2.* The differences 
in columns 8-10 are the errors made by the observers. What strikes the eye here is 
the smallness of the errors and the absence of any systematic character ; the signs 
+ and — are rather evenly distributed. One abnormal deviation of 46' occurs in 
Timocharis' result for Arcturus, where a mistake might be presumed ; otherwise his 
errors do not exceed 15', Hipparchus' largest error is 18', and Ptolemy's largest 28'. 
The mean error for Hipparchus is 7' only, for Ptolemy it is 13', and for the ancient 
Alexandrians Timocharis and Aristyllus it is 14' (or by exclusion of Arcturus 9'). 
As far as a conclusion can be based on this small amount of material — we have no 

* P. V. Neugebauer Sterntafeln von 4000 vor Chr. bis zur Qegenwart, Leipzig, 1912. Inconsistencies between the commuted 
Z^^^J&XS^^ eff6CtS ^ ^ Pr ° Per Ul0D8 - U ^ bC ^ed^T^Va^eTS 



64 



Ptolemy's precession 



other — the excellence of Hipparchus' work is seen to surpass that of both former and 
later Greek astronomers. 

Returning now to the problem of Ptolemy's value for the precession, the question 
may be asked : when this set of observations is so consistent with the modern true 
data, how could Ptolemy find therein a confirmation of his far too small constant of 
precession ? The answer is given by column 7 of Table 1 , containing the errors of the 

Table 2. Comparison of declinations 



Star 


Declinations observed 




Declinations 


computed 




Errors of observation 


Tim.- 


At. 


Hipp. 


Ptol. 


- 288 


- 128 


+ 137 


Tim.-Ar. 


Hipp. 


Ptol. 


1. Altair 


+ 5° 


48' 


+ 5° 


48' 


+ 5° 50' 


+ 5° 


40' 


+ 5° 


41' 


+ 5° 


47' 


+ 8' 


+ 


7' 


+ 3' 


2. tj Tauri 


+ 14 


30 


+ 15 


10 


+ 16 15 


+ 14 


31 


+ 15 


20 


+ 16 


38 


- 1 


— 


10 


- 23 


3. Al debar an 


+ 8 


45 


+ 9 


45 


+ 11 


+ 9 





+ 9 


42 


+ 10 


48 


- 15 


+ 


3 


+ 12 


4. Capella 


+ 40 





+ 40 


24 


+ 41 10 


+ 39 


46 


+ 40 


26 


+ 41 


27 


+ 14 


— 


2 


- 17 


5. Bellatrix . 


+ 1 


12 


+ 1 


48 


+ 2 30 


+ 1 


15 


+ 1 


47 


+ 2 


39 


- 3 


+ 


1 


- 9 


6. Betelgeuse 


+ 3 


50 


+ 4 


20 


+ 5 15 


+ 3 


48 


+ 4 


15 


+ 4 


57 


+ 2 


+ 


5 


+ 18 


7. Sirius 


- 16 


20 


- 16 





- 15 45 


- 16 


12 


- 16 


4 


- 15 


53 


- 8 


+ 


4 


+ 8 


8. Pollux 


+ 33 





+ 33 


10 


+ 33 24 


+ 33 


4 


+ 33 


15 


+ 33 


28 


- 4 


— 


5 


- 4 


9. Castor 


+ 30 





+ 30 





+ 30 10 


+ 29 


58 


+ 30 


5 


+ 30 


11 


+ 2 


— 


5 


- 1 


10. Regulus . 


+ 21 


20 


+ 20 


40 


+ 19 50 


+ 21 


8 


+ 20 


40 


+ 19 


50 


+ 12 


+ 








11. Spica 


+ 1 


24 


+ o 


36 


- 30 


+ 1 


26 


+ o 


33 


- 


56 


- 2 


+ 


3 


+ 26 


12. tj Ursae 


+ 61 


30 


+ 60 


45 


+ 59 40 


+ 61 


35 


+ 60 


41 


+ 59 


12 


- 5 


+ 


4 


+ 28 


13. t Ursae . 


+ 67 


15 


+ 66 


30 


+ 65 


+ 67 


30 


+ 66 


37 


+ 65 


6 


- 15 


— 


7 


- 6 


14. g Ursae 


+ 68 


30 


+ 67 


36 


+ 66 15 


+ 68 


37 


+ 67 


45 


+ 66 


18 


- 7 


— 


9 


- 3 


15. Arcturus . 


+ 31 


30 


+ 31 





+ 29 50 


+ 32 


16 


+ 31 


18 


+ 29 


43 


- 46 


— 


18 


+ 7 


16. a Librae . 


- 5 





- 5 


36 


- 7 10 


- 4 


46 


- 5 


38 


- 7 


4 


- 14 


+ 


2 


- 6 


17. /? Librae . 


+ 1 


12 


+ o 


24 


- 1 


+ 1 


5 


+ o 


15 


- 1 


7 


+ 7 


+ 


9 


+ 7 


18. Antares . 


- 18 


20 


- 19 





- 20 15 


- 18 


22 


- 19 


6 


-20 


17 


+ 2 


+ 


6 


+ 2 



changes of declination between Hipparchus and himself. The six stars selected by 
Ptolemy (indicated by an asterisk in Table 1) are those which present strongly 
negative errors, thus exhibiting too small a change of declination. If we derive the 
precession from these six stars alone we find 2A(5 = 354'; 2 cos a = 5-194, 265w 
= 68-2; n — 0-257 = 15"4; precessional constant = 38", whereas the remaining 
twelve stars give 2A<5 = 645', S cos a = 7-003, 265w = 92^0, n = 0-348 = 21-0, 
precessional constant = 52". 

There can be no doubt that Ptolemy selected these six stars because they were 
favourable to his assumed value of the precession and could be quoted as confirma- 
tions, and that other stars were omitted because they did not confirm his assumption. 
Yet we cannot speak of an attempt to deceive his readers ; he presents to them the 
full material with the unfavourable cases also. It comes down to saying : "my result 
is confirmed by a number of data ; the other data which do not conform to it do not 
count". 

In the third part of the same chapter Ptolemy gives "a still more clear demonstra- 
tion of the phenomena" by determining the change in longitude of some stars. From 
two occultations of the Pleiades by the Moon, one observed by Timocharis in 283 B.C. 
and the other by Agrippa in Bithynia a.d. 92, he finds the longitudes 29° 30' and 
33° 15' and hence an increase in longitude of 3° 45' in 375 years. From three occulta- 
tions of Spica, two observed by Timocharis in 294 B.C. and 283 B.C., and one by 
Menelaus at Rome in a.d. 98, he finds the longitudes 172° 20', 172° 30', and 176° 15', 
giving an increase of 3° 45' in 379 years and 3° 55' in 391 years. From two occultations 



A. Pannekoek 65 

of /? Scorpii, one seen by Timocharis in 295 B.C. and one by Menelaus in a.d. 98, he 
finds the longitudes 212° and 215° 55', and hence an increase of 3° 55' in 391 years. 
The real increase in these cases must have been between 5 and 5| degrees. 

Of this derivation Dreyer says: "A worse selection of material for the object in 
view than the seven conjunctions of the Moon with stars it would be difficult to 
conceive". Because the star places here are based upon Ptolemy's lunar theory with 
all its imperfections they may contain grave errors. Dreyer finds the errors of the 
older and the later longitude to be : for rj Tauri — 73' and + 14', for Spica — 20' 
and + 70', and for /S Scorpii — 41' and + 49'. "By a curious piece of ill luck the 
longitudes for the time of Timocharis were all too great and those for the end of the 
first century too small, which produced the faulty precession of 1° in 100 years". 
Was it ill luck indeed ? Certainly it must be deemed highly improbable that three 
cases, as the only available ones, should by mere chance all be affected by the same 
large error. Considering his dealing with the declinations we must assume rather that 
here also Ptolemy had at his disposal more abundant material of other star occupa- 
tions which he deemed it useless to exhibit in detail since their testimony, not fitting 
in his theory, was of no account. Selection of the data in this way is, of course, 
strictly condemned by modern scientific standards. In condemning Ptolemy we 
should not forget, however, that the principle of selecting data and rejecting deviating 
results as unreliable was followed up to almost modern times ; not until the seven- 
teenth and eighteenth century did it become habitual to derive and use the average 
of all the observed data. Even in the nineteenth century, scientists felt themselves 
warranted in excluding strongly deviating values, and they established an exact 
criterion for exclusion. It is well known that Ptolemy has been harshly criticized 
and suspected to have falsified or even invented the observations used in his deduc- 
tions, e.g. in a detailed way by Delambre (1817).* When at the same time he is 
praised as one of the greatest scientists of antiquity, it is clear that his work cannot be 
judged by the present standards of scientific research. His way of looking at the 
phenomena and their explanation by theory was different from ours ; and this holds 
for the scientists and philosophers of antiquity at large. 

A few cases of ancient eclipses excepted, it is only rarely that the study of ancient 
astronomy can contribute to our present knowledge of the celestial motions. Rather, 
conversely, the numerical data derived from modern research are so accurate that 
by computing the then positions of the celestial bodies, we can examine the accuracy 
of the statements made by the ancient astronomers. Thus in studying the history of 
astronomy we are not learning something about the stars, but something about the 
astronomers. What we are informed about is the psychology of the ancient scientists ; 
we are confronted with what was in their minds, and not only with their opinions on 
the stars and the world, but also with their logic and their mode of thinking and 
arguing in establishing facts and theories. 

Ptolemy, as the author of the most extensive textbook on Greek astronomy, is the 
most prominent object of such study. His prominence appears, for example, in the 
fact that he was the only one among the later authors who took notice of Hipparchus' 
discovery of precession. Struck by the statement, in cautious form, that there was a 
regular displacement certainly as large as 1° per century and perhaps larger, he tried 
to determine it himself in a direct way. His determination by means of Regulus, 

* J. B. J. Delambre, Histoire de VAstronomie Ancienne, Tome I, pp. XXV, XXXI, XXXII; Tome II, pp. 250, 599. (1817). 



86 Ptolemy's Precession 

through the chance concurrence of systematic errors, afforded him the same numerical 
value. Not content with this single result he looked for confirmation by other 
methods ; and in a list of old and new measurements he found six stars whose motions 
agreed with the value derived. Should he not have been cautious, seeing other stars 
that disagreed ? It is a common experience in all later scientific progress that often 
in the first enthusiasm of a new discovery the disagreeing facts are simply dis- 
regarded. Science was not built up as we do it in our textbooks, where the materials 
are given and we know exactly what is relevant and what is not. In early times, 
when nature was so little understood and there was such abundance of unknown, 
mysterious forces in the world, it did not seem surprising that so many cases did not 
fit ; the happy surprise that some few cases did fit well, and therefore appeared more 
important, was sufficient to give confidence that theory was right. Considered in this 
light, Ptolemy's treatment of the precession may be helpful in understanding the 
character of scientific thinking in antiquity. 



A Medieval Footnote to Ptolemaic Precession 

Derek J. Price 

Christ's College, Cambridge 

It is interesting that when Ptolemaic astronomy returned to Europe via Islam and 
Spain the false value of precession was corrected to a remarkably accurate one. The 
Alfonsine astronomers considered the motion of the equinoxes to be compounded 
from a steady and an oscillatory part. The steady motion took 49,000 years for a 
complete rotation and corresponded to the difference between the tropical and the 
calendar years; the modern Gregorian correction omits three intercalary days in 
four centuries, or 365|- days in 48,700 years. The Alfonsine value is equivalent to an 
error in date of equinox of 7 hr in 1000 years — a very commendable accuracy for 
thirteenth-century observations. * 

The oscillatory part of precession, the trepidation, had to account for the residual 
displacement of the fixed stars with respect to the tabulated places of the Sun. It 
was considered to go through i 9° with a period of 7000 years and a zero on 17th May 
a.d. 15. At the date of the Alfonsine Tables, a.d. 1272, the trepidation was 8° 7' and 
the calendrical precession in the same interval of 1257 years from the zero was 9° 12'. 
The total Alfonsine precession was therefore only 11' less than the amount 17° 30' 
calculated from modern tables, and well within the limits of medieval accuracy in 
observation. Unfortunately the trepidation was then approaching its maximum, 
and within a few centuries it became evident that the anticipated decrease in rate 
of precession had not occurred. 

Possibly Ptolemy was led astray by inaccurate values for calendrical precession. 
At all events the authority of his incorrect constant of precession probably led the 
Alfonsine astronomers to believe they had detected a secular variation of that 
constant and permitted them to introduce the erroneous and misleading notion of 
trepidation. 

* A fuller account of Alfonsine precession theory is given by D. J. Price, The Equatorie of the Planetis, Cambridge, 1955, 
pp. 104-7. 



The Peking Observatory in A.D. 1280 and the 
Development of the Equatorial Mounting 

Joseph Needham 

Caius College, Cambridge, England 
Summary 

2?L° f the f ° caI P° ints i n ^e intercourse between civilizations was the year + 1267, when the Persian 
a ronom^rs oTth'e oW™ 7" "»* Y^Dkhan fr0m the M&raghah Observatory to confer with he 
Yu^n SZltn T V k % Y fJ B ^ directed ^ KtJO Shou-Ching. The dynastic history of the 
mm and Tthe in«f ^^V^ ^^ ° T m ° dels of ^truments which the Persian brought with 

"sTmSfied In ^l^T" ^"^^ Shou -Ching "t up about the same time. It is suggested that the 
K^ V ^^^^\-rr tlaS1 ?^ nt10 ^ With the e q uat °™l mounting of modern telescopes, was 
the torou^fm ft wT ^nT™ ° f ^ e J, a l heT mediaeval Arabic and European instrument known as 
Sance S, ?£' ill f simplified" because the ecliptic components had been removed, in accor- 

after the time oY^obT^ C °-° rdmates ' classica11 ^ Chinese > and ad o P ted generally in the West 



Of the armillary spheres of the late Sung dynasty little is known, but some of the 
+ 13th-century instruments of that great scientist and engineer Kuo Shou-Ching [l] 1 
are still extant, 2 and kept at the present time at the Purple Mountain Observatory 
of Academia Sinica, north-east of Nanking (Figs. 1, 2, and 2a). They were still in 
use at the time of the arrival of the Jesuits about + 1600. Here is what Matthew 
Kicci wrote about them 3 : 

''Not only in Peking, but in this capital also (Nanking) there is a College of Chinese 
Mathematicians, and this one certainly is more distinguished by the vastness of its buildings 
than by the skill of its astronomers. They have little talent and less learning, and do nothing 
beyond the preparation of almanacs on the rules of calculation made by the ancients— 
and when it chances that events do not agree with their computations they assert that 
what they had computed was the regular course of things, and that the aberrant conduct of 
the stars was a prognostic from heaven about something that was going to happen on 
earth This something they made out according to their fancy, and so spread a veil over 
their blunders. These gentlemen did not much trust Fr. Matteo, fearing, no doubt, lest 
he should put them to shame; but when at last they were freed from this apprehension 
they came and amicably visited him in the hope of learning something from him. And 
when he went to return their visit he saw something that really was new and beyond his 
expectation. 

There is a high hill at one side of the city, but still within the walls. 4 On the top there is 
an ample terrace, capitally adapted for astronomical observation, and surrounded by 
magnificent buildings which form the residences of the astronomers. ... On this terrace 
are to be seen astronomical instruments of cast metal, well worthy of inspection whether for 
size or for beauty, and we certainly had never seen or read of anything in Europe like them. 
For nearly 250 years they have stood thus exposed to the rain, the snow, and all other 
atmospheric inclemencies, yet they have lost nothing of their original lustre. . . . 

First we inspected a great (celestial) globe, graduated with meridians and parallels; 
we estimated that three men would hardly be able to embrace its girth. ... A second 
instrument was a great (armillary) sphere, not less in diameter than that measure of the 

* Th g ™i? ? qUa -? b £ ac J<?£ s refer t0 the table of Chinese characters on p. 68 

been publteLd b^ R MtoER Potsdam. Good photographs of them taken during their period at Potsdam have 

* ]fe™t toS'A 2 ^ 315 - Q» ot fd by TRiaAULT (1615), tr. Gallagher, p. 329 if., and by Yule 
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outstretched euros which is commonly culled tt geometric pace, It had a horizon (-circle) 
and poles; instead of circles if was provided with certain double hoops (arinillae), the void 
space between the pair .serving the Mimic purpose us the eirelcs of our spheres/' AH these 
were divided into ;Wi> degrees and some odd minutes. There was no globe representing the 
earth in the centre, but there was a certain tube, bored like agim-batTel. which could readily 
be turned about and Mxrd to any ,'iziniuth or altitude so as to observe any particular star, 
just as wo do with our vane-sights — not al all a despicable device. . . . 

The third machine was a gnomon, the height of which was twice the diameter of the 
former instrument, erected on a very large and long slab of marble, on the northern side of 
the terrace. The stone slab had a channel cut round the margin. In be filled with water in 
order to determine whether the slab was level or not, and the style was set vertical as in 
hour dials. We may suppose this gnomon to have been erected that by its aid the shadow 
at the solstices and equinoxes might be precisely noted, for in that view both the slab and 
the style were gra dilated. 




Fig. 1. Armjllaiv spline (eijuattiruil) of Kuo Slum Hiing. ulmut I -7<i (made for the 

latitude of Vhing-Yang in Shansi. hui later ttt Peking and now at Nanking), The photo- 
graph is one by Sain mills which YxTLB obtained from Wvui:. Another good one is in 

Thomson, vol. iv 



The fourth and last instrument, and the Largest of all, was one consisting ol, as it were, 
:; or I huge astrolabes in juxtaposition ; each of Ibcm having a diameter of such a geo- 
metrical pace as I have specified. The fidueial line, 0* Alhidnda. as it is called, was not 
lacking, nor yet the Dioptwt. Of these astrolabes, one having a tilted position in the direc- 
tion of the south, represented the equator; a second, which stood crosswise OH the lirsf, 
in a north and south plane, the Father took Tor a meridian, but it could be turned round on 
its axis; a third stood in the meridian plane with its axis perpendicular, and seemed to 
represent a vertical circle, but this also could be turned round to show any vertical whatever. 
Moreover, all these were graduated and the degrees marked by prominent studs of h-on, so 
that in the night the graduation could he read by touch without any light- All this com- 
pound astrolabe instrument was erected on a level marble platform with channels round it 
for levelling. 



" This was to iilluw of tin' HwhiKliiK oJ'IIlc wlgWtag-fcalM WltMn Uu'sudrcHlur Bttidai It 1> noteworthy that ,iin-lt & device Wus 
strange to lticci. 



70 Tho Peking Observatory in a. it. 1280 rand the development of tin- equatorial mounting 





Kig. ->, "Simplified instrument" (really tin equatorial torquetum) of Kim shon-fliing, 
about 127(1 {made fin- tin- latitude of 1'liing-Ynng in Shansi, but later at Peking and 
now tit Nanking). The photograph is one by Saxthders which Yi 1.1; obtained from U'vi.ik; 

it was taken at Nanking 



Joseph Nkedhaji 



71 




Fig. 2 («). bee Fig. 2: a mora reoml photograph of the "Simplified Instrument" (taken at 
Nanking) showing the "pole-deter mining circle" «t the top, with its crossbars and central hole. 

attached to the "normal cirel©*' 

The table of identifications given below refer* to the sketch (in the lower part of the opposite page, 
Fig. 2. 



a, a North pole cloud frame standards (pei chi yiin ehia f "1*5 T) . 

b Xormal circle (kuei huan [30]). diameter 2 ft 4 in, (fixed). 

c', e,' Dragon pillars (lung elm (_3"« J )- 

d, d South pole cloud frame standards {nan chi yiin ehia |SS]J, 

e (fixed). Diurnal circle (pal kho huau [89]}, graduated in the twelve bout's and the hundred 

divisions"] each division containing 3<j snb-divisioii.s; it carries four roltefl for the better 

rotation of the equatorial circle. Dian inter (\ ft 4- in. 
/ (revolving). Meridian, double circle (sail yii shuang huan | K)|), <> ft in diameter, graduated in 

degrees and minutes. For declination. 
if, (/' Stretchers (c-hih chii [41]). 
h Brace (heng [42]). 
i Diametral alidade (khuei heng |43J), with pointed ends (like ceremonial tablets) for accuracy 

(tuan wci kuei shou [43aJ), and sighting-vanes (heng erh | 43b]). 
,/ (revolving). Equatorial circle (ehhih tao huan | 44 1), o ft in diameter, graduated in degrees and 

minutes. For right ascension. 
k, k' 'Independently movable radial pointers with pointed ends (ohJah heng [45]). It is not olear 

wliether these carried sighting vanes. Their name ("boundary bars 1 ') indicates that they were 

used to mark off the boundaries of the hsiu |4,"m|. i.r, the equatorial lunar mansions. 
J. Pole- deter mining circle (ting chj huan \4U\). Diameter equivalent to 0°. Attached to the 

upper part of the normal circle. It cannot be seen in Fig- - hnl appears clearly in big. 2 (a). 

This circle had a cross-piece inside it, with a central hole, and seems-; in have been used for 

determining the moment of culmination of the pole-star itaolf. Observation was effected 

through a small bole in a bronze plate attached to the youth pole cloud frame: standards. 

The main polar axis through the centre of the diurnal and normal circles was also pn>\ idi-d 

with holes constituting a sighting-tube. 
m (fixed). 'Earthly co-ordinate circle (yin wei huau |47|), for azimuths. 
it (revolving). Vertical circle with alidade (I.i villi huan [48 |). for altitudes. 



72 The Peking Observatory in a.d. 1280 and the development of the equatorial mounting 



On each of the instruments explanations of everything were given in Chinese characters, 
and there were also engraved the 24 zodiacal constellations which answer to our 12 signs, 
2 to each. 6 There was, however, one error common to all the instruments, i.e. that in all 
the elevation of the pole was taken to be 36°. Now there can be no question about the fact 
that the city of Nanking lies in lat. 32 |°, whence it would seem probable that these instru- 
ments were made for another locality and had been erected at Nanking, without reference 
to its position, by someone ill- versed in mathematical science. 7 

Some years afterwards Father Matteo saw similar instruments at Peking, or rather the 
same instruments, so exactly alike were they, insomuch that they had unquestionably been 
made by the same artists. And indeed it is known that they were cast at the period when 
the Tartars were dominant in China; so that we may without rashness conjecture that they 
were the work of some foreigner acquainted with our studies." 8 

Thus Ricci was greatly impressed by the astronomical instruments of the Yuan 
dynasty, though holding a poor opinion of those Chinese contemporaries whom it 
was his strategy to supplant, and venturing a particularly erroneous guess about 
the original constructor of the instruments. 

The authentic Chinese texts from which we gain information about the equipment 
of Kuo Shou-Ching's observatory of + 1276 include, of course, the Yuan Shih 
(History of the Yuan Dynasty). 9 The instruments are there listed as follows: 



(1) Ling Lung I [2] 



(2) Chien I [4] 

(3) Hun Thien Hsiang [5] 

(4) Yang I [6] . 

(5) Kao Piao [8] 

(6) Li Yiin I [9] 

(7) Cheng Li I [10] . 



Ingenious Armillary Sphere (Fig. 1). 

Ricci's "second instrument". The instrument of this kind 
still preserved at the Purple Mountain Observatory near- 
Nanking is considered to be the copy made by Huangftj 
Chung-Ho [3] in + 1437, and not the original one of 
Kuo himself. 

Simplified Instrument (Fig. 2, 2a). 
Ricci's "fourth instrument". 10 

Celestial Globe. 

Ricci's "first instrument". 

Upward-looking Instrument. 11 

A hemispherical sun-dial intermediate in size between 
earlier Chinese types and the much larger Jai Prakas 
instruments of the Indians. 

Lofty Gnomon. 12 

Undoubtedly the 40 ft gnomon, especially that at Yang- 
chhSng. 13 

Vertical Revolving Circle. 

The vertical circle in (2) described by Ricci, and seen in 
Fig. 2. This, with (13) below, would be equivalent to 
modern altazimuths and theodolites. (Spencer Jones (1), 
p. 83.) 

Verification Instrument. 

It is not clear what this was, but perhaps it was a com- 
ponent of (1) which permitted exact determinations of the 
positions of the sun and moon near the equator; such at 
any rate was its stated purpose. Perhaps a sighting-tube. 



6 This was a mistake of Ricci's. 

7 This point can be cleared up at once. The instruments which Ricci saw belonged to a set which had originally been 
made for the astronomical college founded by the great statesman and astronomer Yehlu Chhu-Tshai [la] about + 1220 at 
Phingyang in Shansi (latitude just over 36°). In the scientific decadence of the Ming they had been removed to Nanking 
(Johnson). That the astronomers of Kuo Shou-Ching's time were very conscious of the importance of the latitude is shown by 
the fact that the Yuan Shih (Chapter 48, p. 12b, ff) reproduces a table of the latitudes of some twenty-five important centres, 
some of which undoubtedly had astronomical instruments; it is entitled "Ssu Hai Tsche Yen" [7]. See Gaubil, p. 110. 

8 Tr. Yule, i, p. 451; cit. Bernard-Maitre, p. 59; Wyiie, p. 14. 

9 Chs. 48 and 164, paraphrased and partly translated by Wylie, who lists the other sources. 

10 Yuan Shih, Chapter 48, p. 2b. 

11 P. 6a. 12 P. 8b. 

13 Ricci's "third instrument" must have been one of the usual 8-ft gnomons. 



Joseph Xkkdham 



73 



(H) Chlng Pta [11] 
(9) Khuei Chi [12] 



Shadow Definar. 14 A device for determining more precisely 

the end of the sun shadow on (3). 
Observing Table. Apparently an adaptation of the gnomon 

and shadow-definer to lunar shadows. 
















m 



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i iS'vV^.t- \o Mm 5 * s\ -n*?« *■' 

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6 5? * __£b 



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P%, 3. Annillary Sphere of Su Sung (ft — 1070/ - I0!»f>). us described ii-i his hook Jfe&J i 
Ssiang Fa Yfto (Now Description of the ArmilWy Sphere) of ■• 1004 (from thv s/iuu Shan 

Ko Tfihuny Shu edition) 



(10) Jib Yiicb tthih I £IS] . Instrument for Observation of Solar and Lunar Eclipses. 

Not clear and not explained. 
"'Star-dial". Could this have hern a forerunner of tin- - Kith 

century nocturnal "t 
Time-determining Instrument. Probably tin- ''diurnal circle" 

of {'2) above- 
I Hi* ct ton-determining Table. This must have been an azi- 
niuthal circle, probably the "earthly co-ordinate circle" 
of (2) above. 



(11) Using Kuci[l4] . 

(12) TingShih I [16] . 
(IS) Cheng Fang An flO] 



" P. iiu. 



74 The Peking Observatory in a.d. 1280 and the development of the equatorial mounting 

(14) Hou Chi I [17] . . Pole-observing Instrument. 

Presumably a quadrant for obtaining the polar altitude 
and hence the latitude. Or it may have been the polar 
sighting-tube embodied in (2) above. 

(15) Chiu Piao Hsiian [18] . Nine Suspended Indicators. 

Though details are not given, this probably refers to the 
"groma" or hanging plumb-lines whereby the trueness of 
the instruments, especially the gnomons, was checked. 

(16) Cheng I [19] . . Rectifying Instrument. 

Purpose uncertain. 

Apart from the minor devices and those about which we have not sufficient infor- 
mation, the chief interest lies in the great armillary sphere (1), and the "Simplified 
Instrument" (2). There is little need to describe the former more fully, since it did 
not differ in any fundamental way from that which Su Sung [19a] had already used 
in + 1090 (Fig. 3), though no doubt of finer and exacter workmanship. But the 
second of these two, which Ricci could only describe as a collection of "astrolabes" 
set in different axes, is something new. I believe that it was a variant of the mediaeval 
instrument known as the "torquetum" 15 which consisted of a series of discs and circles 
not, like those of the armillary sphere, placed concentrically. The Arabic and 
European versions had a disc mounted in the plane of the equator, and another one 
revolving at any angle to it in the plane of the ecliptic ; this then carried a celestial 
longitude circle at right angles to itself 16 . The equipment was completed by a half- 
disc or protractor and plumb-line for reading off altitude. Probably the apparatus 
was used mainly for computations, as it permitted the direct conversion of ecliptic 
to equatorial co-ordinates and vice versa as well as other comparisons. The invention 
of this unwieldy instrument has often been ascribed to Kuo Shou-Ching's elder 
contemporary, Nasir al-Din al-Tusi of Maraghah 17 but more probably goes back 
to the Spanish Muslim, Jabir ibn Aflah (b. about + 1130). 18 Europeans were 
using it at the same time as the Maraghah astronomers in Persia (Thorndike, 1, 2) 
and there were treatises on the torquetum by Regiomontanus and Apianus about 
+ 1540, 19 but Tycho Brahe spoke scornfully of it, 20 and after him no one employed 
it except the Indians who perpetuated the Arabic tradition. An outdoor example is 
still extant at Jaipur, the Krantivritti Valaya Yantra 21 (see Fig. 4). 

Now the great point of interest about Kuo Shou-Ching's "simplified instrument" 
is that although (as a "dissected" armillary sphere) it is recognizably related to the 
torquetum, it is a true equatorial. Doubtless it was because the ecliptic components 
had been removed that the instrument received its name "simplified". This means 
that though Arabic influence may have been responsible for suggesting its construc- 
tion, Kuo adapted it to the specific character of Chinese astronomy, namely 
equatorial co-ordinates. And in so doing, he fully anticipated the equatorial mount- 
ing so widely used for modern telescopes. This is generally attributed to James 
Short, F.R.S. (+ 1710/+ 1768) in + 1749. 22 Here we reach the situation which 
several recent historians of science have recognized as being of such great interest, 

15 To be distinguished carefully from the "triquetrum", see on. 

" See R. Wolf, ii, p. 117; Hottzeau, p. 952; Gunther (1), vol. 2, pp. 35, 36; Anon., p. 18, No. 348 and opp. p. 30; 
Rohde, pp. 79 if; Michel (2), p. 68 and PL XIII. One of the best expositions is that of Michel (3), who showed that the 
Rectangulus of Wallingford, which Gunther (1), vol. 2, p. 32, had not understood, was a skeleton torquetum. 

17 Sarton, ii, p. 1005. 1S Sarton, ii, p. 206. 

18 The oldest European specimen is that preserved at Cues near Treves; Nicholas of Cusa bought it in + 1444 (see Hartmann.) 

20 Baeder, Stromgren, and Stromgren tr., p. 53. 

21 Kaye, pp. 32, 33, and Fig. 58; Soonawala, p. 38. This instrument now lacks several components. 
82 Cf. Chauvenet, p. 367, who favours Fraunhofer; and A. Wolf, pp. 136 fF. 



Joseph Nredham 



namely («) the faithfulness of the Chinese from the beginning of their history to 
what afterwards became the co-ordinates universally used, and {&) the influences 
which may have acted upon Tyi.'ho Bk.vkk to cause- him to abandon the character- 
istic Graeco- Arabic- European system of ecliptic co-ordinates. 

Before examining this, however, it will be more logical to explore somewhat further 
what is known of the Arabic influence upon the astronomers of the Yuan dynasty.' 23 
The task is easy owing to the special studies which Hartner; Yabucchi; and 
Tasaka 24 have devoted to the subject. They have been able to Identify the diagrams 
of seven astronomical instruments which reached China from Persia in + L2(>7, In 




Fi£. 4. The only existing outdoor tor-quotum, the KriinLivritti Vnlnyit Yantra at the 
Jaipur Observatory (from Kayk) 

the YtmnShik^ a couple of pages are devoted to the Plans (or Models) of Astro- 
nomical Instruments from the Western Countries" (Hsi yii i hsiang [20|) — these 
were sent by Hulagu Kuan or his successor to EJEUBXLA1 Khan through the hands of 
one of the Maraghah astronomers 0ha-Ma~1jB-TbsQ | 2 1 \ 2i] (Jam II AL- 1 )Is) in | 1 267. 
The identity of this man is somewhat obscure but he may have been the JamIl 
al-Din iBN Mu.rrAMWAD a ^-Najjar!. 27 who bad declined to take on the full responsi- 
bility of building the Maraghah observatory in -f 12o8. The Chinese names of the 
instruments about which he was deputed to inform the Chinese astronomers, together 
with brief explanations of them, are given in the Yimn Shih text. 
Those instruments were the following: 

Persian- Arabic Chinese translation and 

Chinese Iranwrvptitm 11 * ariainat. ex/pltmoM&n 

(I) Tsii-lhu ha-la-chi [22 | Dhaka al-haluq I Inn Tlm-n I |23] 

{"the owner of tlu> Annillnry sphere, 
rings"). 

■ u niNKB \\m described two iatensttne manuscripts preserved hi Ut tibat is line ftdkawo Observators in Etoeate, one in 
Amide or Persian, tin's other In Chinese. Thev are tables of the motion oF.-om, moon, snd planets, calculated from an epoch 
sturtiiiK nt i i2iit,und written abort — 1BB1. M probable rettc* of the collaboration of Jah.U vt.-nistmd Kro SHor-Cmso, 
they would be preetonB Indeed, and it is to be hoped that r]n -y wore not dcstmywl uiien th>> observatory was Injrnt ihiriii^ 

W'nrlil Wur H. Ell Flic following ecimirv, there were it erous Arablc-Chliii'-ic scientific contauta, 

1 < i Ki tiis. J n the followliiK identifications of Arabic terms and their Interpretations we urrnpt those of Hartsee rather 
than thorn! of Tasaka. 

■■• Chapter 4w, ]». lob, ** Cf. Samto.v. ii, p. 1021. 

« Or more probably al-BukharI {note from Dr. W r Hartmer). 

! " These are mil the imlj extant traiisrrlptions. but they are the most correct. CliluVti-bntig edition* id" it"- Vuaa S/uh were 
Hiibjeeted to revision by a commission of learned linguist* wbti uniiiKolielsu'i] alt forclitn word* even when these hail been trans- 
llteratlon* from other lanKunKe* Boon as Arabic, (note from l)r, W. IIaki SKk). 



76 



The Peking Observatory in a.d. 1280 and the development of the equatorial mounting 



(2) Tsa-thu shuo-pa-thai [24] 



Dhatu'sh shu'batai 
("the instrument with 
two legs"). 



(3) Lu-ha-ma-i 
miao-wa-chih [26] 

(4) Lu-ha-ma-i 
mu-ssu-tha-yii [28] 

(5) Khu-lai-i sa-ma [30] 



(6) Khu-lai-i a-erh-tzu [32] 

(7) Wu-su-tu-erh-la [34] 



Rukhama-i-mu f wajja 

Rukhama-i-mustawiya 

Kura i-sama f 

Kura i-ard 
al-Usturlab 



Tshe Yen Chou Thien Hsing Yao 
chih Chhi [25] ("instrument for 
observing and measuring the 
rays of the stars of the celestial 
vault"). Certainly Ptolemy's 
"organon parallacticon", 'opyccvov 
n&poLXXoLKTiKOv, i.e. the long ruler, 
or "triquetrum", for determining 
zenith distances of stars at 
culmination. 29 

The conjectures of Zinnbr, p. 
236, that it was a divided circle, 
and of Fuchs, p. 4, that it was the 
Jacob's staff of the surveyors, are 
not to be retained. 
Tung Hsia Chih Kuei [27] 
"Solsticial Dial", i.e. plane sun- 
dial for unequal hours. 
Chhun Chhiu Fen Kuei [29] 
"Equinoctial Dial", i.e. plane 
sun-dial for equal hours. 
Hsieh Wan Hun Thien Thu [31] 
("obliquely set globe with map of 
the stars"). Celestial Globe. 
Ti Li Chih [33] 
Terrestrial Globe. 
Astrolabe. The text says: "The 
Chinese name (for this) has not 
been worked out. The instru- 
ment is to be made from bronze, 
on which the times (hours) of the 
day and night are engraved". 
Certainly not a clepsydra, as 
Zinneb, p. 236, supposed. 

The list is an interesting one. The first suggestion was certainly no novelty for the 
Chinese, 30 but Jamal al-Din's instrument was surely adapted for ecliptic measure- 
ments, and as we saw above, Kuo Shoit-Ching paid no attention to this. It would 
also have used a graduation of 360° but Kuo Shou-Ching retained the 365^° system. 31 
Nor was the fifth instrument, the celestial globe, anything new. On the other hand, 
the terrestrial globe was perhaps a novelty ; there is no previous record of one before 
the time of Martin Behaim (+ 1492) 32 except the ancient globe of Crates of Mallos 33 
in the — 2nd century, which had been entirely forgotten. The Chinese text describes 
the new instrument as "a globe to be made of wood, upon which seven parts of water 
are represented in green, three parts of land in white, with rivers, lakes, etc. Small 
squares are marked out so as to make it possible to reckon the sizes of the regions 
and the distances along roads". There seems no evidence, however, that the Chinese 
took this up. As for the sun-dial, they were probably puzzled at the conception 

28 The best description is Tycho Bkahe's (Baeder, StrSmgren, and Stromgren, p. 44). See also Gunther (1), ii, p. 15. 

30 Hartner goes far astray in his suggestion that armillary spheres were unknown in China until the + 13th century — an 
under-estimate of perhaps seventeen centuries. We have details of some twenty-five important instruments from the — 2nd 
century onwards, and good reasons for thinking that the astronomers of the — 4th also used them, at any rate in simple form . 
This is said not in any criticism of Hartner, since, as he informs us, his facilities were lamentably inadequate, and the work of 
Maspero (1, 2) was not known to him; but in order to correct views which might otherwise claim the authority of I sis in 1950. 

31 This seems the only justification, and not at all a strong one, for Johnson's emphasis on what he calls the "tragic con- 
servatism" of the Chinese. Bosmans, himself a Jesuit, has on the contrary drawn attention to the imposition of the sexagesimal 
graduation of degrees and minutes on the Chinese by Verbiest at a later date. The Chinese had always graduated them deci- 
mally in tenths and hundredths. Bosmans freely admits that this change was a retrograde one. 

** Ravenstein. 3S Sarton, il85; cf. Stevenson; Schlachter and Gisinger 



Joseph Needham 77 

of unequal hour's, and it is fairly clear that their traditional type of sun-dial persisted 
unchanged. 

The instrument most strikingly absent from the list is the torquetum, though it 
would have been expected more than any other, if Kuo Shou-Ching's "Simplified 
Instrument" arose from the stimulus of contact with Arabic science as on all the 
circumstantial evidence it did. Moreover, no torquetum is listed in al-'Urdi's 
account of the equipment at the Maraghah Observatory (Seemann (1); Jourdain). 

As for the second and seventh of the instruments, if they were not adopted it was 
surely because they did not fit into the characteristic system of Chinese astronomy, 
polar and equatorial. The parallactic ruler for determination of zenith distance 3 * 
could hardly interest astronomers in whose work the zenith played no part. And the 
astrolabe, so universal in Arabic and mediaeval European astronomy, was primarily 
intended to measure altitude, and to compute ecliptic co-ordinate positions, which 
the Chinese did not particularly want. Hartner considers that what the Maraghah 
astronomers offered was well chosen ; 35 they did not send the apparatus for determining 
sines, azimuths, and versed sines, 36 because it was probably known that the Chinese 
astronomers were unfamiliar with spherical trigonometry. Jamal al-DIn had an 
overwhelming task before him if he intended to explain to the Chinese the whole 
system of Arabic gnomonics and the mathematics required for the stereographic 
projection on which the astrolabe markings were based, and if he tried he certainly 
did not succeed. But what has been unperceived, even by Hartner, is that the 
measurements and computations which the rulers and the astrolabe could yield were 
simp]y not wanted in the Chinese polar-equatorial system. 

The astrolabe is a very complex instrument upon which mediaeval Arabic and 
European astronomers lavished all their mathematical art. It might be called a 
"flattened" armillary sphere, 37 combining the armillary rings of Hipparchus and 
the theodolite of Theon with a projection of the zodiac and the starry hemisphere. 
The luminous treatise of Michel (1), published a few years ago, makes recourse to 
older explanations of the theory and construction of astrolabes unnecessary, while 
a massive compendium by Gunther (2) gives elaborate details of the principal 
surviving instruments. The place of origin of the instrument is uncertain, 38 but its 
first known user was the Byzantine Ammonios (ca. + 500), though the earliest 
dated astrolabe is Persian, that of "Ahmad and Mahmud the sons of Ibrahim the 
astrolabist of Ispahan", + 984. Venerable also is the Byzantine astrolabe of + 1062, 
described by Dalton. The earliest extant treatise on its use derives from Joannes 
Philiponus (+ 525), the Byzantine physicist, a pupil of Ammonios, and in the 
next century the Syrian bishop Severus Sebokht also wrote on it. 39 Not a single 
example is known from China, either by textual references or actual preservation. 

It was Dreyer (1) who was perhaps the first to appreciate the historical importance 
of Kuo Shou-Ching's retention of the equatorial system in his Simplified Instrument. 
"We have here", he said, "two remarkable instances of how the Chinese people 
often came into possession of great inventions many centuries before the western 

34 Illustration in Gunther (1), ii, p. 15. 

36 It will be remembered that the Maraghah Observatory had on its staff at least one Chinese astronomer. His name was 
apparently Fu Meng-Chi, or as some think, Fu Mu-Chai. Cf. Sarton, vol 2 p 1005 

36 Cf. Jourdain and Seemann. 
rJL H J, n a C e ^lP% me " astrol fbium planisphaerium", R. Wolf, ii, 45. There was an intermediate form, in which flat rings were 
&£?»£* %££££&? IfiXif^l ^ tWS ^ n6Ver widespread - 0ne such > made ^r Ileonso X, is figured^ 
„™f£(™ E ,? GEB ^ UEI V- Wha * Ptolemy calIe a an astrolabe was an armillary sphere, a fact which has caused some terminological 
co *'°„ ,I , n m °dem times Ricci's use of the word astrolabe in our opening quotation was, of course, quite unjustified 
Gunther (1), vol. v. SeealL J.VrTnk astronomer Manasseh of Baghdad (d. + 815) (Messahalla) has been translated in 



78 The Peking Observatory in a.u. 1280 and the development of the equatorial mounting 

nations enjoyed thorn. Wo find here in the 4- 13th century the equatorial armillae 
of Tycho Brahk, and better still, an equatorial instrument like those 'armillae 
aequatoriae inaximae' with which Tycho observed the comet of -f- 1585 as also the 
fixed stars and planets"/ 10 

JOHNSON considers that "the Yuan instruments exhibit a simplicity which ia not 
primitive, but implies a practised skill in economy of effort, and in this .sense compares 
favourably with the Graeco-Muslim tendency to rely on separate instruments for 
each single co-ordinate to be measured — neither Alexandria nor Maraghah 41 exhibit 




Kig. 5. Tycho UkaICk's Armilln Aeqtmtoria Maxima 
(from H. Hakivkh, E, Sthoiiuhkn. ana B. &VS&UGBMR) 

any device so complete and effective and yet so simple as the 'simplified' instrument of 
Kuo Shou-Chtng. Actually our present-day equatorial mounting has made no further 
essential advance". 42 Johnson adds that the gun-barrel sighting-tubes rotating in 
the double rings were much preferable to the open alidades of the Arabs, 13 

An urgent question thus arises. What was it that led Tycho Bkahk in the 
-f 16th century to abandon the age-old Graeeo-Muslim ecliptic co-ordinates and 
ecliptic armillary spheres in favour of the equatorial co-ordinates which the Chinese 
had had all along t Gunther 44 expressed the greatest surprise that the Chinese had 

*" Si-i! Viu. 'k The iliffrrorKv was Ihiil XYcHu ri-hiiuiJil u half-tirrlc uf Hi iiuitur hrm-inu I he lumr-ciri-li- rmtrullv. 

11 For Jiatu ortlio .Unrfiftfuili equipment see .Iouuiain itrul skemank (I), sutumariml in sartos, if, HIIU. 

" it is inti"r«>stlii(j Id ri'lioet that- Ma net) i'or.o W41* In China jimt at the Hum when all thi* iwanoliii(oii, lint hi- nnllied un[v 
tin; astrological as|ioc(s of Chinese Htiitc-HiipporW a*r<< r*>Ti<im>-, Some iifliis fi-xi* contain » elinplt-r (Clutptei ::." m-<; i'l'i.k 
(I J, I. p. 44U), •■CiiiiiMTniiiK I hi 1 AHtrolojji'Mh) tin: City <if Ciimhnlttc", Mi-hiv* Mini no h-tw than £>{)(K> af them were cniirliiined 
liy t In- (in :i Khiin " tli ;in jiimmil iiiflintumuK-n .mil dulhinu, ami (hut "Ihoy have a kind of astrolabe mi whlrh jire bttorfbad 
rlu- planetary siHim, tin- hours und rrilU-nl poinU of the whole yenr". All three seels of Htar-ilerkt*, (.'wlluiyuis, BaHMJen, and 
Christian (presumably rintmr Nesiorlan), used tlicsc hii*trumeiitn for proKiionlliittlons, i>n which lh«'V wen: (-tins ill led l>v musiv 
people, Moreover, they prepared "rortMn little pamphlet* called Tanihr, Lt. Ar, Taqwlm, calendars or ophemeridfift, which 
were ])u1iJIhIk:i1 by this government in surprising numbers. 1'ur example, in I :i2s mure Maui Miree million roples were printed 
and issued, Sdinc of these (<v/, one for 1 l4iP*)]ftt<:r<-iimt- Into I hi' possession <il'mieh mm :i> Kohkht Hhvi.k, Koheut Kohikk, 
and Samukl, I'kpys, amiiHing t I k-1 r curiosity roiiccrnlntt t'hinesc astronomy. 

" Another mutter in which the Kenuissance astronomers of Kin-ope adopted Chinese practices vena the (tre alcr use of meridian 
transit oliservaMoiw. There wfiH Tvciio'a (irent iiinrnl <|iiadr:ml. followed hv \\w Iti'^r inuiiiiriim »i'n i.']i-..i].i- in-nii:un-nilv in 

tin' tiii'lMhnj by liOKMKK ill ] |QX| (llHKYKIl (I), tiKANT, |)|t, Ml ff, ; SPBN'l'BH JOKB8 (2)). 

" (1), II, (jp, 145 ff, 



■JOSEPH XkHJJHAM 



used equatorial armillae, which were considered one of the chief advances of European 
Kenaissance astronomy, and concluded that Kuo Siiou-Omsa had anticipated 
Tvcho by three centuries. Now Tycho tells us himself in his book on insirumeiit- 
construetion ,Fl that he found zodiacal or ecliptic armillary spheres very unsatisfactory 




Fig. (). Tv ■oho Brahk's smaller equatorial armillai'v sphere 
(from K. Kaeder, E. Stromqrest, and B. Sthomoiikm) 



because since they are not always in a position of equilibrium (i.e. since their centre 
of gravity shifts according to the position of the equinoctial junctions) they become 
deformed owing to the weight of the metal, introducing errors of. as much as a couple 
of minutes of are. For this reason, he preferred to construct equatorial armillary 
spheres (Fig. <>). But a purely technical reason of this kind seems insufficient for a 
change in the basic method of expressing celestial co-ordinates. DEBTOR (2) there- 
fore raised the question of whether there had been some Arabic influence sapping the 
assured foundations of ecliptic methods. Sedillot 1 ™ (1) suggested that equatorial 
armillary spheres had been known to the Arabs, and ho provided more evidence 
later (2), quoting, among other sources, Bettim's seventeenth-century view that 
al-Haitham had used them in the Jate -j- 10th or early -f- 1 1th century. 47 In such a 
place as this it is impossible to pursue the matter further, but its importance would 

" Tr. It ikder, BtrOhgke.v. and S-nuiMGRBW, pp. "i5 ff". " 1'. 198. 

■ l\ axxxft. BamRl (Apikrta, U. prcgyin. lit, p.. 4t) had written: "Ailhilmit 't'Vfiio iirniillan- i[n.ixlriiim liistruiAenium 
ijutxl tjiini'ii i-umjieri ego positiun et aiiliiliitum oliai fnbw*- anle TvrHi) uli Alhiim-no" 



Sfl 



The Poking Observatory En a.ik 12S0 and the development, of the equatorial mounting 







Si> 






fe 




^ 2 



£ ari 



II £3 



« H " d| 
r*> ^ "^ ^ 



J&i s 



Joseph Needham 81 

seem to justify a special investigation. There exists the possibility that in spite of the 
general trend of Muslim astronomy, there were isolated instances of the use of 
equatorials, and that these might have been derived from Arab-Chinese contacts 
(at any time indeed back to the Han). 48 The idea might then have stimulated a few 
European astronomers, such as Gemma Frisius, who in + 1534 first described a 
small portable equatorial armillary sphere, 49 and then in turn Tycho Brahe himself, 
whose reasons for making the change can surely hardly have been confined to the 
purely technical ones mentioned above. 

How paradoxical it is, in the light of all this, that when the Jesuits proceeded to 
enlighten the Chinese in scientific matters, they erected (-{- 1673/+ 1714) ecliptic 
armillary spheres at the Peking observatory 50 (Figs. 7 51 and 8 52 ). And Verbiest, in his 
book of + 1687 about astronomy in China, had nothing to say about the Yuan 
instruments, save that they had been the products of a "ruder Muse". 

48 If we may include pre-Muslim times and the Syrian and Persian predecessors of Arabic science. 

49 Drbyer (2), p. 316. 

60 Lbcomte, p. 67. They are still in good condition (1952) and have retained their sighting-tubes (square-sectioned outsido 
and tubular inside) with fiducial cross- wires. 

51 Other good photographs were published by B. F. Robinson. 52 From Verbiest, copied by Lecomte and others. 

Note 

This study, which it has given the author much pleasure to contribute to a collection 
honouring Professor F. J. M. Stratton, will form part of the third volume of a work 
Science and Civilisation in China (now in course of publication by the Cambridge 
University Press), the first volume of which appeared in the autumn of 1954. 

He wishes to thank Mr. Wang Ling and Dr. Derek Price for valuable assistance. 



References 



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Muller, Amsterdam), 1911. 
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1935. 

Bettini, M Apiaria (Bononia), 1645. 

Bosmans, H "Une Particularity de l'Astronomie Chinoise au XVIIeme 

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Chatjvenet, W Manual of Spherical and Practical Astronomy (Lippincott, 

Philadelphia), 1900. 
Dalton, O. M "The Byzantine Astrolabe at Brescia [of a.d. 1062]", Proc. 

Brit. Acad., 1926. 
Dreyer, J. L. E. (1) "The Instruments in the Old Observatory in Peking", Proc. Boy. 

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Century (Black, Edinburgh), 1890. 
Frank, J Die Verwendung des Astrolabs nach al-Chwarizmi (Mencke, 

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Fuchs, W The 'Mongol Atlas' of China by Chu Ssu-Pen and the Kuang Yu 

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Mathematiques, Astronomiques, Geographiques, Chronologiques, 

et Physiques, tirees des anciens Livres Chinois, oufaites nouvelle- 

ment aux Indes, a la Chine, et ailleurs, par les Peres de la Com- 

pagnie de Jesus, ed. E. Souciet (Rollin, Paris), 1732, vol. ii. 
Grant, R History of Physical Astronomy , from the Earliest Ages to the Middle 

of the XlXth Century (Baldwin, London), 1852. 
Gunther, R. T. . . . . . . Early Science in Oxford, 14 vols. (Oxford), 1923-45, especially 

vols, ii and v. 
Gunther, R. T The Astrolabes of the World, 2 vols. (Oxford University Press, 

Oxford), 1932." 

7 



82 The Peking Observatory in a.d. 1280 and the development of the equatorial mounting 

Hartmann, J "Die astronomischen Instrumente des Kardinals Nikolaus 

Cusanus", Abhdl. d. Gesellsch. d. Wiss. z. Gottingen (Math.-Phys. 
KL), 1919 (NF) 10, No. 6. 

Hartner, W "The Astronomical Instruments of Cha-Ma-Lu-Ting, their 

Identification, and their Relation to the Instruments of the 
Observatory of Maragha". Isis, 1950, 41, 184. 

Houzeaix, J. C Vade Mecum de V Astronomic (Hayez, Brussels), 1882. 

Johnson, M. C "Greek, Muslim, and Chinese Instrument Design in the Sur- 
viving Mongol Equatorials of 1279 a.d.", Isis, 1940-47, 32, 27. 
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towards a Modern Revision of their Antagonism (Faber & 
Faber, London), 1944. 

Joxjrdain, A "Memoire sur l'Observatoire de Meragha et les Instruments 

Employes pour y Observer", Magasin Encyclopedique, 1809 
(6eme ser), 84, 43; and sep., Paris, 1810. 

Kaye, G. R The Astronomical Observatories of Jai Singh (Government 

Printing Office, Calcutta), 1918 (Archaeol. Survey of India, 
New Imp. Ser., No. 40). 

Lecomte, Louis "Memoirs and Observations Topographical, Physical, Mathema- 
tical, Mechanical, Natural, Civil, and Ecclesiastical, made in a 
late Journey through the Empire of China, and published in 
several Letters, particularly upon the Chinese Pottery and 
Varnishing, the Silk and other Manufactures, the Pearl 
Fishing, the History of Plants and Animals, etc., translated 
from the Paris edition, etc.", London, 1698. 

Maspero, H. (1) "L'Astronomie Chinoise avant les Han", Toung Pao, 1929, 

26, 267. 

(2) "Les Instruments Astronomiques des Chinois au Temps des Han", 

Melanges Chinois et Bouddhiques (Bruxelles), 1939, 6, 183. 

Michel, H. (1) Traite de V Astrolabe (Gauthier-Villars, Paris), 1947. 

(2) ...... Introduction a VEtude d'une Collection d 'Instruments Anciens de 

Mathematiques (de Sikkel, Antwerp), 1939. 

(3) "Le Rectangulus de Wallingford, precede d'une Note sur le 

Torquetum", Ciel et Terre, 1944, (Nos. 11 and 12), 1. 

Muller, R "Die Astronomischen Instrumente des Kaisers von China in 

Potsdam", Atlantis, 1931, 120. 

Neugebauer, O "The Early History of the Astrolabe", Isis, 1949, 40, 240. 

Raeder, H., Stromgren, E. and 

Stromgren, B Tycho Brake's Description of his Instruments and Scientific Work, 

as given in his "Astronomiae Instauratae Mechanica (Wandes- 
burgi, 1598), (Munksgaard, Copenhagen), 1946. (Publication 
of K. Danske Videnskab. Selskab). 

Ravenstein, E. G Martin Behaim; his Life and his[Terrestrial] Globe (London), 1908. 

Robinson, B. F "The Astronomical Observatory in Peking", Art and Archaeology 

(Washington), 1930. 

Rohde, A Die Geschichte d. wissenschaftlichen Instrumente vom Beginn der 

Renaissance bis zum Ausgang des 18. Jahrhunderts (Klinkhardt 
& Biermann, Leipzig), 1923. 

Sarton, George Introduction to the History of Science, 5 vols. (Carnegie Institution 

of Washington, Washington D.C.), 1927-47 (Publication No. 
376). 

Schlachter, A. and Gisinger, F. . Der Globus, seine Entstehung und Verwendung in der Antike 

(Teubner, Leipzig and Berlin), 1927. (Stoicheia, Stud. z. 
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Seemann, H. J. (1) "Die Instrumente der Sternwarte zu Maragha nach den Mit- 

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Maharaja Sawai Jai Singh II of Jaipur and his Observatories 

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136, 349. 



Singer, C. and Singer, D. W 
Soonawala, M. F. ... 



Spencer Jones, H. (1) 

(2) 



Joseph Needham 83 

Stevenson, E. L Terrestrial and Celestial Globes; their History and Construction, 

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additional note by J. L. E. Dreyer), Copernicus, 1882, 2, 123. 

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Wolf, R Handbuch d. Astronomie, ihrer Oeschichte und Literatur, 2 vols. 

(Schulthess, Zurich), 1890. 

Wylie, A "The Mongol Astronomical Instruments in Peking" (Travaux 

de la Illeme Congress des Orientalistes, 1876) ; incorporated in 
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(Springer, Berlin), 1931. 



The Mercury Horoscope of Marcantonio Michiel of Venice 

A Study in the History of Renaissance Astrology and Astronomy 
Willy Hartner 

Institut fur Geschichte der Naturwissenschaften, 
Frankfurt a.M., Germany 

Summary 

A statuette of Mercury (1527) wrought by Antonio Minelli and consecrated by a Venetian patrician, 
Marcantonio Michiel, carries a horoscope combined with a graphic demonstration of the planet 
Mercury's motion according to the Ptolemaic system. 

The first part of the study deals with the dating and astrological significance of the horoscope proper. 
It is shown that a uniquely propitious moment: 15th June, 1527, 8 a.m., was chosen to mark some 
important event in Michael's life. Judging from the general character of the horoscope, it is likely that 
it was destined to mark the conception of Michiel's first child. As his son Vettore was born only 7 months 
later, the possibility of a premature birth must be envisaged. The concluding note (at end of study) 
evidences that the data of the horoscope were directly taken from Johannes Stoffler's ephemerides 
for the years 1499-1531. 

The second part of the paper discusses the history and development of Ptolemy's theory of Mercury. 
In a digression the algebraic curve described by the centre of Mercury's epicycle is analysed. It is shown 
that this curve is practically interchangeable with an ellipse. The curve appears for the first time in a 
treatise by Azarquiel (eleventh century) which is preserved in a Spanish translation incorporated in the 
Libros del Saber. European historians, with the exception of A. Wegener (1905), have failed to recognize 
the true significance of this curve, which is by no means an anticipation of Kepler's ellipses though it 
may have been one of the stimuli that led to his experimenting with oval (elliptiform) curves. G. Peur- 
bach (fifteenth century) almost totally depends on the Islamic astronomers' (Azarquiel, Alhazen) 
interpretation of the Ptolemaic theory; he deals in extenso also with the elliptiform curve. It is finally 
shown that the geometrical designs filling the interior of the horoscope were directly copied from 
Peurbach's Theoricae Novae Planetarum. 



1. Introduction 

Some three years ago, the Victoria and Albert Museum in London was presented, 
by Dr. W. L. Hlldburgh, F.S.A., with a marble statuette of the god Mercury, 1 
carved by a sixteenth-century Venetian sculptor, Antonio Minelli (Figs. 1 and 2). 
From a study by John Pope-Hennessy 2 we learn that little is known about the life 
and work of Minelli, whose fame ranks lower than that of his father, Giovanni, of 
Padua. Antonio's name first occurs in a contract signed in the sacristy of the Santo 
at Padua, on 21st June, 1500. In 1525 he went to Venice, where a few of his works 
are preserved. There is no other evidence about him, and even the year of his death 
remains unknown. 

If we content ourselves, for the moment, with this brief statement on the place held 
by Minelli's Mercury statuette in the history of cinquecento art, we may turn to 
one peculiarity which makes it remarkable from an astronomical point of view. On 
the left-hand side of the figure, the artist has placed a small altar bearing on its inner 
side an inscription which refers to the completion and dedication of the statuette ; 
it reads as follows : 

MERC. SIMVLAC. 

M. ANT. MICHAEL. P.V. 

ANN. VRB. VENET. 

MC. VI.F.C. 
CONSECRAVITQ. 

1 No. A 41 — 1951. 

2 A statuette by Antonio Minelli, in The Burlington Magazine, January, 1952, pp. 24-28. 

84 



Wxvus Hammer 



85 





Fig. 1. Side- view or inarbJe statuette, showing 
the bronze plaque 



Vig. 2, Olhn sj«|i> of libs imirhle statuette, 
showing tlie insn iptiim {imt visible on photograph) 



ANT, SIT NELL. SCVLI'T. 
VATAVIXVS. 

XVI. KAL. MARTI. OOBFTVM. 

XVII. KAL. QVIXT. 
PE R FECIT 

ANN. SAL. 
M.D.XXVII. 



On the two narrow sides of the altar appear a crane and a boarded, flat-nosed, 
horned deity, probably representing Pan {the son of Hermes-Mercury and the 
nymph Dry ope) whose characteristic features are frequently confused with those of 
Hilcnus, another of Hermes 1 descendants, On the outer side (under a relief where 
the eaduceus of Mercury in the centre is franked on the left by a lyre placed over a 
cook, and on the right by a knapsack 3 over a long-legged bird, apparently also a 
crane), a circular bronze plaque with an engraved horoscope is suspended: see 
Fig. 3. Tt is this horoscope with which we are principally concerned. 4 

The "horoscope" as a whole is of particular interest because, contrary to the 
normal type, it stresses the predominance of one single planet. Mercury, over all of 
his companions. This is proved, on the one hand, by its combination with the 
Mercury statuette and the various Mercurial emblems; on the other, by the fact that 
the interior of the plaque gives a complete illustration of I he mathematical theory of 



* Not a acfir;Moii. u hrterjpffetetl i>v pope-hknxfksy. The lyw and the kimin+aik ure the tw> Indl&pGas&ble attributes of 
Mercury, the god of poets, scholars. and merchants (including thieves). The COCK ami the erase :i!*'> frequently decor la com- 
bination with Mercury, tt wottH take tow long to go into B diBCttSsbu nf rln-lr < nluin . Tln>y mv commonly hitrniivti'.i as 
symbols oi vigilance and learning* 

* Owing to the kindness of my" friend Dr. Arthtr Beer, and tliruuirh Iiih kuoiI (rittirs, a photograph of tin 1 horoscope was 
aetit me by the WA.HBiKii In'stititk of the University <>r London. The nwnti o!' my roEiJinittuion from i In 1 ihtta of the horo- 
scope (dttte 1">rh June, 1527) was found to agree with the Inscription cited; eee POPB-BBKHTBBBY, Ijt. p. 2&. 



86 The Mercury Horoscope of Marcantonio Michiel of Venice 

Mercury, as conceived in early Hellenistic times and perfected by Ptolemy and 
Islamic astronomers, representing the extravagant motion of the most capricious of 
all the seven planets. 

1. ASTROLOGICAL PART 

2. The Horoscope Proper 

The plaque has a diameter of 102 mm. In Fig. 4 all its characteristic details are found 
represented on an enlarged scale. The outer rim, consisting of three circular rings, 
carries the ordinary division of the zodiac into twelve signs of 30° each, starting 
from the vernal point, shown on the right half of the horizontal diameter, and 
running counter-clockwise from Aries to Pisces. The longitudes of six of the seven 
planets as well as that of the ascending node of the Moon, are indicated by the usual 
planetary symbols followed by Roman numerals. Only the position of Mercury is not 
listed. This apparent omission, however, is readily explained by the circumstance 
that the position of Mercury can be obtained by sighting from the centre of the 
horoscope over that of the disk of Mercury in the upper half of the horoscope, and 
reading off the graduation of the rim. It yields 2° Cnc == 92°. Thus we have the 
following positions : 

Sun = Jupiter . 2° Cnc = 92° 

Mercury . . 2° Cnc = 92° 

Moon . . .20° Cap = 290° 

Venus . . .26° Cnc = 116° 

Mars . . .23° Sco = 233° 

Jupiter (see Sun) 

Saturn . . . l°Tau= 31° 

Ascending Node . 24° Sgr = 264° 

As is evident from these figures, our horoscope presents the unusual case of a 
simultaneous conjunction of the Sun with Mercury and Jupiter in 2° Cnc, Venus 
standing 24° east in the same sign, and the Moon — about 35 h after her full — close to 
opposition with Venus. Mars, holding his "domicile" Scorpio, has passed his 
opposition with Saturn, but the two ill-omened planets still stand in diametrically 
opposite signs, whereby their disastrous influence, according to most contemporary 
astrological theories, is increased. On the other hand, the danger of the situation 
may be considered extenuated by a favourable coincidence : Saturn stands in the 
auspicious sextile aspect with the Sun, Jupiter, and Mercury; and Venus, in the 
equally favourable trigonal aspect with Mars ; secondly, the Ascending Node, no less 
evil than Mars and Saturn, holds the sign of Sagittarius which astrologers regard as 
its "dejectio", i.e. the place where its action is minimized. 5 More will have to be 
said about the astrological significance of the horoscope in connection with an 
attempt at clearing up its history (see below, Sections 5-7). 



6 For further information, see Willy Hartnek, "The Pseudoplanetary Nodes of the Moon's Orbit in Hindu and Islamic 
Iconographies", in Ars Islamica, Vol. V, Pt. 2, pp. 114-154, Ann Arbor, 1939 ; and A. Bouche-Leclercq, L'Astrologie Orecgue, 
Paris, 1899, Ch. VII, pp. 180-240. 



Willy Hartner 



87 



3. The Date oe the Hoeoscope 

The computation of the date is most conveniently effected with the aid of P. V. 
Neugebatjer's tables. 6 The longitude of the Sun indicates a day about the middle of 
June (in the period 1200-1600, the date corresponding to a mean solar longitude of 
92° recedes from 17th June to 13th June, Jul. style 7 ). On the other hand, a rough 
calculation of the positions of the slow planets, Jupiter and Saturn, and of the 
Ascending Node (period of revolution 18-6 years) immediately points to the summer 
of 1527. Thus we have to look for a date in the middle (close to the 15th) of June, 
1527, which lies about 35 hours after the true full-moon. According to NT, II, 83-88, 
the mean full-moon of June, 1527, occurred on 14th June, about 2 h a.m. (G.M.T.), 
and the true opposition less than l h later, the mean anomaly of the Moon being close 
to 0°. By adding 35 h , we arrive at the date of 15th June, 1527, about l h p.m. Hence 
it is this day, if any at all, that fulfills the conditions defined by the horoscope. 
Evidently, however, the hour of the day as found by our rough calculation, cannot be 
expected to be correct, for neither do the Moon- tables in use during the sixteenth cen- 
tury warrant an exactness to the degree, nor can a rough estimate like the one applied 
here be regarded as conclusive. But there is another way of determining the hour with 
a higher degree of accuracy, viz. by computing it from the longitude of the ascending 
point of the ecliptic (see below, Section 5). Anticipating that the result is about 
8 h a.m. 8 , the theoretical true geocentric longitudes of the seven planets and the 
Ascending Node are given below as computed from NT, II, for the date 15th June, 
1527, 8 h a.m. The second line indicates the data derived from the horoscope, the 
third, the difference "Horoscope minus computed" 9 . 

The congruence is indeed remarkably good. It not only excludes any doubt of the 
correctness of our date, but also proves that the horoscope was cast by a skilled 
astronomer, and probably with the aid of the best astronomical ephemerides or tables 





Sun 


Moon 


Mercury 


Venus 


Computed 
Horoscope 


92?40 
92 


287?57 10 - 11 
290 


89?33 12 
92 


116?44 13 
116 


Horoscope — Computed . 


0° 


+ 2° 


+ 3° 


0° 






Mars 


Jupiter 


Saturn 


Ascending 
Node 


Computed 
Horoscope 


232?08 
233 


92?76 u 
92 


28?83 J1 
31 


264? 1 
264 


Horoscope — Computed . 


+ 1° 


- 1° 


+ 2° 


0° 



6 Tafeln zur Astronomischen Chronologie, Vol. II: Sonne, Planeten und Mond, Leipzig, 1914 (cited "NT, II"); III: Hilfs- 
tafeln zur Berechnung von Himmelserscheinungen, 2nd ed., 1925 (cited "NT, III"); and Astronomische Chronologie, Vols. I-II, 
Berlin and Leipzig, 1929. 

' All dates are given in Julian style. 

8 Evidently, in this context, the exact hour of the day is relevant only in the case of the Moon. 

* Thus in analogy with the modern "O — C". The data of the horoscope were definitely not observed but computed in 
advance or taken from ephemerides then in use. See Concluding Notes at the end of the present study, p. 135. 

10 Reduced to the meridian of Venice (X = + 12°20). 

11 In the case of the Moon, Jupiter, and Saturn, perturbations in the mean heliocentric longitudes as well as in the eccen- 
tricity and the longitude of the perihelion, were taken into account. Their effect is practically negligible. For the Moon, the 
main perturbations in longitude are comprised in the position indicated. 

18 l-2 d after inferior conjunction. 

13 60-61 d after first appearance as evening star. 



88 The Mercury Horoscope of Marcantonio Michiel of Venice 

obtainable. It seems likely that these were either Johannes Stoffler's ephemerides 
for the years 1499-1531 (Almanack nova plurimis annis Venturis inservientia, Ulm, 
1499) or else the Alphonsine Tables 14 , the fame of which had not yet been eclipsed 
by those compiled at the time immediately after the death of Copernicus : Erasmus 
Reinhold's Tabulae Prutenicae ("Prussian Tables") and the tables of Tycho 
Brahe. But even these new tables were so far from perfect that it is dubious whether 
they really marked a decisive advance on the Alphonsine Tables. One repeatedly 
encounters complaints from the contemporary astronomers about the unreliability 
of the Prussian Tables which were frequently found to give less accurate results than 
Tycho Brahe 's and even the Alphonsine Tables; thus Kepler states that, in the 
case of the planet Mars, their deviation from the positions observed in 1625 amounts 
to 4°, or even 5° 15 . Unfortunately, Stoffler's ephemerides as well as the Alphonsine 
Tables, which played a predominant part in the history of astronomy of the fourteenth 
to the seventeenth century (according to Zinner 16 they were printed in ten editions, 
1488-1649, and besides, hundreds of handwritten copies are preserved to-day), were 
inaccessible to me so that I was unable to verify their assumed connection with the 
horoscope 17 . 

4. Origin and Purpose of the Horoscope. The Inscription on the Altar 

The date derived from the horoscope by purely astronomical means is fully con- 
firmed by the second part of the inscription incised on the back of the altar (see 
Section 1), which reads in translation: "The sculptor Antonio Minelli of Padua 
finished the work, which he had started on 14th February, on 15th June, 1527 A. D." 18 . 
In other words, the date for which the horoscope was cast coincides with the day 
on which the artist put the last hand to his work. This obviously implies that 
not only the statuette, but also the plaque carrying the horoscope had been com- 
missioned in advance to be ready on the day indicated on both, which means that the 
horoscope must have been precalculated in order to celebrate one single day of 
decisive importance in the future life of the person who commissioned the statue. 

The possibility of an ordinary birth horoscope thus being excluded, the question 
arises as to what may have been the purpose of working out a horoscope valid for a 
day in the near future — about four months in advance. It is well to remember that 
precalculated horoscopes of this kind were hardly less frequent than the ordinary 
ex post facto specimens like birth horoscopes. A Persian king, in the Islamic Middle 
Ages, would hardly have dared to mount a horse without previously having consulted 
his court astrologer, and still in the seventeenth century (Thirty Years' War, Wallen- 
stein) the dates of battles could depend just as much on astrological forecasts as on 
purely strategic needs. During the Renaissance it also became customary for private 
persons of rank and importance to follow the example of statesmen and generals, in 
order to know what steps to take and which to avoid. Hence we may safely assume 
that Marcantonio 's horoscope either had an apotropaic significance, viz. serving 



" Or perhaps the tables of Giovanni Bianchini (ca. 1458), based on the Alphonsine tables, and reduced to the meridian of 
ierrara. See R. Wolf, Oeschichte der Astronomie, Munich, 1877, p. 79, and E. Zinner, Geschichte der Sternkunde, Berlin, 
1931, p. 371. 

" £ f - E - Z I NNBR . l - c -> P- 462. 16 I.e., pp. 369-70. " See "Concluding Notes" at the end of this study (Section 13). 
Concerning the first part of the inscription, which gives the commissioning and consecration date "1106 Urbis Venetorum' 



by me) hasta la eleccion del dux Enrico Dandolo, lo que pudiera llamarse periodo de desarrollo (421-1192) . . ". The 
foundation date indicated there (421 a.d.) tallies perfectly with the inscription on the altar: 421 + 1106 = 1527. 



Willy Hartner 89 

the purpose of averting threatening danger, or that it was destined to mark a day on 
which the fates were peculiarly propitious. 

It is understandable that Marcantonio Michiel — one of the central figures of 
Venetian humanism and, as a patrician of his home town, undoubtedly a man of 
considerable wealth — should have chosen the planet Mercury as his protector and 
regulated his life according to the caprices of his astrological patron. Unfortunately, 
however, although we possess a study on the life and works of Marcantonio 19 , 
our knowledge does not suffice to allow of a definite answer to our problem; but 
the little we happen to know about Marcantonio 's private life during the years 
1527 and 1528 does throw some light on the significance of his horoscope. 

From a playful remark in a letter to Sadoleto, written by Girolamo Negro in 
1 527, we learn that Marcantonio in that same year had married a girl without dowry : 
Ceteri communes amici bene habent praeter M. Michaelum nostrum qui uxorem duxit, 
quamquam in hoc quoque philosophum egit, indotatam enim accepit, ne, ut est Plautinum 
Mud, dote imperium venderet. The statement is confirmed by the Genealogie del Bar- 
baro 20 according to which the marriage of Marcantonio to Maria Soranzo took 
place in February, 1527. Recalling to our memory that the work of the Mercury 
statuette was begun on 14th February of the same year and anticipating that there is 
sufficient astrological evidence to assume that the marriage was celebrated at about 
the same time (see below, Section 8), it seems more than a mere guess to postulate a 
connection between the two events. Indeed, there is a fairly high probability of the 
horoscope's bearing directly upon the married life of the young couple. The first 
thing to be thought of, then, would be that it is in some way connected with Marcan- 
tonio's eldest child, Vettore, who was born on 13th January, 1528. 21 But in this case 
no other interpretation appears to offer itself than that the horoscope was set for a 
day particularly auspicious to conception. Conception horoscopes are known to have 
been quite a common phenomenon from the days of late antiquity down to the period 
concerned. Therefore, it would hardly surprise us to learn that Marcantonio, 
obviously a firm believer in astrology, had allowed himself to be guided by astrological 
considerations even in the most intimate steps of his private life. 22 However, counting 
back from the day of Vettore's birth by an average normal period of 280 days, would 
bring us down to 8th April, 1527, and for the probable date of conception to about 
15th April, which is two months earlier than indicated by the horoscope. The 
possibility of a premature birth might be thought to explain the discrepancy ; 23 if 
new documentary material on Vettore should bear witness to an unusual weakness or 
untimely death of the child this might support our hypothesis. On the other hand, of 
course, a second possibility cannot be excluded, namely, that nature prevailed over 
astrological considerations, and that Vettore was conceived six weeks or two months 
before the propitious day. 



19 E. A. Cicogna, "Intomo la vita e le opere di M. Michiel patrizio veneto della prima meta del secolo XVI," in Memorie 
dell' Istitwto Veneto, IX (1861), p. 383 (cited after Pope-Hennessy, not consulted by the writer of this article). 

20 MS. Venice, Museo Correr, Cic. 515, MSS. II, 174, Vol. V. 

21 According to a private information of Dr. Giovanni Mariacher of the Museo Correr, cf. Pope-Hennessy, I.e. p. 25, 
footnote. 

22 I am, of course, aware that the lapse of four months between the marriage and the horoscope date is rather long. However, 
there was no lack of medical devices to comply with the needs of astrology. 

23 According to modern statistics, the average period of post-conceptional pregnancy is 273 d . The actual period can be con- 
siderably shorter, but the probability rate for the birth of a mature child before the 250th day post conceptionem decreases 
rapidly. One out of 4299 children is born before the 246th day, and one out of 3,333,333 before the 234th day. This, of course, 
concerns only the question of maturity, not of viability. The lower limit of viability is assumed to be 181 d post conceptionem, 
which is a good deal less than the period between the day of the horoscope and that of Vettore's birth (212 d ). Yet, even in our 
days, the cases of viable children born after less than 215 days' pregnancy post conceptionem seem extremely rare, and it is 
doubtful whether such a possibility could even be considered in the Renaissance. Cf. L. NUrnberger, "Abnorme Schwanger- 
schaftsdauer", in Biologic und Pathologic des Weibes, Vol. VII, part 1, Berlin and Vienna, 1927, pp. 365-406. 



JJO 



Tho Mercury Horoscope of Mar can ton io Miuhiol of Venice 




.Kig. 3, Dotaiirt of the plaque 



Finally, a third possibility must be examined, namely, that the date of the horo- 
scope is meant to mark the moment of animation of the embryo. Against Tertulltan, 
I)e anima, Chapter XXVI I, who assumes the soul to come into being simultaneously 
with conception, St. Aimustink, Qnucxt. in Exod. quaest, hXXX, distinguishes 
between the unanimat-erl and I lie animaled embryo. Relying on tin's passage, Canon 
Law 24 assumes the male embryo to be animated on the 40th, and the female on the 
HOtli day, whereas Secular Law indiscriminately assumes the 40th day for both sexes. 
But, obviously, there is no reason to believe that the question of animation is appli- 
cable to our case; on the contrary, the fact alone that a definite hour is indicated by 
the horoscope, is sufficient to refute such an interpretation. 



For rei'urvm'L.'*, vS. L. >f UiiNiiKiuncK, "Fehlgebort unrL FrOhgeburt", op. e&, pp. 4(>7-«4fi {am in particular p, 4U), 



Willy Hartneb, 



91 



5. The Hour of the Horoscope. The Ascendant and the Twelve 

Houses (Loci). Domiciles and Exaltations. Aphetic Points and 

Aphets (Hyleg). The Wheel of Fortune 

The relative positions of the planets in the zodiac determine only the astrological 
situation in general. In order to establish a complete forecast of future events, which 




Fig. 4. Marcantonio Michiel's horoscope (redrawn) 



is the true purpose of all horoscopes, the exact knowledge of the hour, and even the 
minute, is indispensable. For only if the day and the hour are known is it possible to 
compute the decisive point of the horoscope, namely, the degree of the ecliptic rising 
above the horizon at the moment concerned, which serves as a starting point of a 
most peculiar, and no less illogical and abstruse, division of the zodiac into twelve 
houses (Lat. loci) of unequal extent. 

In our case, this ascending point, or short, Ascendant (Greek copoaKo-n-os, Lat. pars 
horoscopans, whence "horoscope"), is found by producing the sloping straight line 
(see Figs. 3 and 4) through the centre of the horoscope (marking the horizon of 
Venice) and reading on the rim, in the left upper quadrant, which yields 18° Leo, 
or 138°. The point diametrically opposite, 18° Aqu, or 318°, is called Descendant 
(Lat. occasus) as it sets at the same moment that the Ascendant rises 25 . 



25 Theoretically, the inverse interpretation according to which the ascendant and the descendant would have to be inter- 
changed, is possible. But in horoscopes the ascendant nearly always stands on the left, so that the upper half of the circle 
represents day, the lower, night. Moreover, such an inverse interpretation would not tally with the position of the Moon, nor 
would it lead to any reasonable astrological result — so far as the adjective reasonable can be used in connection with astrology. 



92 The Mercury Horoscope of Marcantonio Michiel of Venice 

(a) The Hour 

The equatorial co-ordinates corresponding to an ecliptical longitude of % = 138° 
are a = 140?5, or 9^36 26 , and 6 = + 15?5. To d = + 15?5 corresponds, for the 
latitude of Venice (O = + 45?43), the semi-diurnal arc t = 7*15. The sidereal time 
at mean noon of 14th June, 1527, is ■& = 6*09 27 . Hence 

a — t — = 20*12 
Correction for mean time = —0-05 



Hour of horoscope T = 20*07 Mean Time Venice, counted from noon, 14th June 
T x = 8 h 4m a.m. Mean Time Venice, 15th June, 1527 28 . 

It is this value of T 1 which we had anticipated in checking the planetary positions 
indicated by our horoscope (see above, Section 3). 

As will be shown later, the reasons for which this moment T x was chosen are 
manifold. Only one of them need be dealt with now because it serves at the same 
time to prove the correctness of our calculation. 

In converting T x into unequal ("temporal") hours, i.e. sixths of the Sun's semi- 
diurnal arc S = 7*83, we obtain (equation of time on 15th June, J = + 0*02): 

Sunrise, 15th June, 1527, Venice, at 12* — S + J 

= 4*17 = 4 h 10 m Mean Time Venice 
$S= 1*305 = l h 18 m 3, hence 

unequal hour No. 



I starts at 4& 10™ 


II starts at 5 


28 


III starts at 6 


47 


IV starts at 8 


5 


V starts at 9 


23 


VI starts at 10 


42 


VII starts at 12 






This means that the moment T x marks the beginning of the 4th unequal hour of 
the day (i.e. after sunrise) of 15th June, 1527 (Saturday). According to the theory of 
chronocratories 29 ("planetary week") each of the 168 hours of the week is governed 
by one of the seven planets which succeed one another in the following sequence : 
Sun, Venus, Mercury, Moon, Saturn, Jupiter, Mars. Starting with the Sun as 
chronocrator of the first hour of the day, on Sunday, the 12th hour of the day 
will be ruled by Saturn, the 1st of the night by Jupiter, the 12th of the night 
by Mercury, and the 1st of the next day (Monday) by the Moon. Similarly, the 
1st hour of Tuesday (dies Martis) belongs to Mars, of Wednesday (dies Mercurii) to 
Mercury, etc., so that each day is initiated by the planet after which it is called. 

28 Fraction of an hour, not minutes. 

27 According to NT III, T. 1 ; hours counted in the old-fashioned way, i.e. from noon of preceding day. 

28 Accuracy of our calculation ± 3 m . As the hour of the horoscope was probably determined by means of an astrolabe (see 
below, Section 6), we have to allow of an additional reading error of ± 1° = ± 4 m . Henee the hour intended, T, is defined by 
7 h 57 m < T <S h ll m . 

" Cf. BOUCHfi-LECLERCQ, op. tit., pp. 476-480. 



Willy Hartner. 93 

Thus, on the day in question, a Saturday, the chronocrator of the 1st hour of the 
day is Saturn, of the 2nd Jupiter, of the 3rd Mars, and of the 4th, which has our 
particular interest, the Sun. 

I venture to say that this is not due to a pure game of chance because, in our 
horoscope, the Sun is practically equal in rank to the predominant planet, Mercury. 
It is the Sun that stands in conjunction with Mercury and Jupiter, marking the 
point second in importance to the Ascendant ; moreover, the Ascendant itself is in 
the sign of Leo, which from remote antiquity had been regarded as the solar 
dominion par excellence, and which scientific astrology, by the beginning of Hellenism, 
attributed to the Sun as his "domicilium". When the Sun stands in Leo, he exercises 
his maximum action. But also by itself, Leo has the qualities of the great luminary 
and plays the role of his deputy. 

(b) The Twelve Houses 

Concerning the method of dividing the ecliptic into twelve houses, the disagreement 
which rules among mediaeval astrologers is remarkable indeed. It would carry us 
too far to enter upon a detailed discussion here. Therefore, referring the reader to 
Bouche-Leclercq's standard work 30 and, particularly, to C. A. Nallino's excellent 
historical summary 31 , we limit ourselves to a few remarks. 

All astrologers agree on the four cardinal points of the horoscope: Ascendant, 
Descendant (Occasus), "Medium Coelum" (abbreviated MC, i.e. the point of the 
zodiac being in upper culmination), and "Imum [Medium] Coelum" (abbreviated 
IMC, i.e. the point in lower culmination). Counting counter-clockwise (with in- 
creasing longitudes), the Ascendant marks the beginning of the 1st house, IMC the 
4th, Occasus the 7th, and MC the 10th. The further subdivision of the four quadrants, 
however, is effected according to methods or hypotheses that differ widely from one 
another; strangely enough, late mediaeval and Renaissance astrologers, instead of 
trying to reconcile conflicting assumptions, even competed in inventing new systems, 
thus adding to the confusion inherited from their predecessors. Yet it appears that 
one system, which will be described here, was considered superior to all others. It is 
the one that Ptolemy seems to have had in mind when dealing with the subject in 
his Tetrabiblos, and which all the later famous Arabic and Jewish astrologers made 
use of in their computations, such as al-Battani (Albategnius), al-Qabisi (Alcabi- 
tius), Abraham ben 'Ezra (Avenezra), Abtt'l-Wafa', and Ultjgh Beg. Through 
Alphons X of Castile's Libros del saber de astronomia and through the writings of 
early Italian astrologers, such as Giovanni Campano's (thirteenth century), it found 
its way also into European astrology and maintained a predominant position during 
the subsequent centuries 32 . 

According to this system, the semidiurnal and the seminocturnal arcs of the 
Ascendant are each divided into three equal parts. Then the sections of the ecliptic 
limited by the great circles laid through the poles and the divisions of the equator 
represent the six eastern houses, running from No. X (which starts at the meridian) 
through XI and XII (ending at the horizon), and on from I (below the horizon) 
through II to III (ending at the northern half of the meridian). 

The six western houses, Nos. IV-IX, are situated symmetrically with the former, 
the point of symmetry being the centre of the Universe (i.e. the centre of the Earth). 

30 Op. cit., Ch. IX (pp. 256-310). 

31 In AL-BattanI, Opus Astronomieum, Part I (Milan, 1903), pp. 246-249. 

32 See "Concluding Notes" at end of study (Section 13). 



94 



The Mercury Horoscope of Marcantonio Michiel of Venice 



Thus we obtain six pairs of diametrically opposite houses each of equal extent 
(No. VII = No. I -f 180°, etc.), but varying in size according to the position of the 
vernal point relative to the horizon 33 . The absurdity of this method could not be 
better characterized than by the words of J.-B. Delambre, in his Histoire de 
Vastronomie du moyen dge M , where he deals with the system of Alcabitius : 



HOR 




occ 



Fig. 5. The twelve astrological houses 
according to Ptolemy's system 



IMC 



"Les six maisons dernieres sont toujours diametralement opposees aux six 
premieres; il en resulte cependant une espece d'absurdite. Le quart de l'equateur, 
entre le meridien et l'horizon occidental, se trouve divise suivant les arcs nocturnes, 
quoiqu'il appartienne au jour; le quart entre l'horizon occidental et le meridien 
inferieur, est divise suivant les heures du jour, quoiqu'il appartienne a la nuit. 
Au reste, le calcul de cette methode est extremement simple, et c'est peut-etre 
ce qui a fait passer sur l'absurdite que nous venons de remarquer". 

In applying the theory of the twelve houses as outlined here, to our special case, 
we arrive at the following result : 

Longitude of Ascendant A = 138°; 

AR of Ascendant a = 140?5; 

Semidurnal arc of Ascendant t d = 7 h 15; 

= 107?25; 

3^ = 35-75; 

Seminocturnal arc t n = 4^85; 

= 72?75 









\t n = 24?25. 




House No. 


I (Ascendant) . 


a = 140?5 


1 = 138?0 = Leo 


18?0 




II . 


164-7 


163-4 = Virgo 


13-4 




Ill . 


189-0 


189-8 = Libra 


9-8 




IV (IMG) . 


213-2 


215-5 = Scorpio 


5-5 




V . 


249-0 


250-6 = Sagittarius 


10-6 




VI . 


284-7 


283-5 = Capricornus 


13-5 




VII (Occasus) 


320-5 


318-0 = Aquarius 


18-0 



33 Twice a day (not once, as erroneously stated by Kallino, I.e., p. 247), the four quadrants of the zodiac will be equal, 
namely, when the vernal or the autumnal point coincides with the horizon. But, evidently, an equality of all of the twelve 
houses can never occur. 

34 Paris, 1819, p. 502. 



Willy Hartneb. 95 



House No. VIII . 


344-7 


343-4 = Pisces 


13-4 


IX . 


9-0 


9-8 = Aries 


9-8 


X (MC) . 


33-2 


35-5 = Taurus 


5-5 


XI . 


69-0 


70-6 = Gemini 


10-6 


XII . 


104-7 


103-5 = Cancer 


13-5 



In Fig. 4, the beginnings ("cusps") of the houses corresponding to the values of 
A indicated, are marked by roman figures (at the outer rim). Fig. 5 represents the 
same houses in a more conventional schematic way as a square subdivided into 
twelve small triangles, leaving a small square open in the middle of the figure. There 
the values of X are marked, in signs and degrees, at the edges of the triangles, while 
the positions of the planets are inscribed inside them. 

The astrological significance of these imaginary "houses" is described by the 
following mediaeval distych : 

Vita, lucrum, fratres, genitor, nati, valetudo. 

Uxor, mors, pietas, regnum, benefactaque, career. 

In other words, House No. I, commonly called Horoscope, decides on the life of the 
newly-born (or the future life of the foetus) in general ; II, on pecuniary circumstances 
("lucre"); III to V, on brothers, parents, and children; VI, on health and disease; 
VII, on marriage ; VIII, on death and inheritances ; IX, on religion (and also travels) ; 
X, on domicile and State (also honours, arts, character, and conduct of life) ; XI 
(Bonus Genius), on benefit, charity, and friends; XII (Malus Genius), on enemies, 
captivity, and other afflictions. 

Not all of the twelve houses are attributed equal weight. The four cardinal ones 
(I, IV, VII, X) are, of course, considered predominant, though IV (IMC) and VII 
(Occasus) are often regarded of lesser interest. Next to these in importance are those 
standing in propitious aspects to the Ascendant, viz. the trigonal (V and IX) and the 
sextile (III and XI). The four remaining (II, VI, VIII, XII) are regarded as "lazy" 
and least effective. 

Remembering that the hour of Marcantonio's horoscope was precalculated, it 
seems astonishing at first sight that the Ascendant was chosen by the astrologer in 
such a way that, with the exception of Mars in the relatively unimportant House 
No. IV, no other planet stands in the cardinal houses. As I am going to demonstrate, 
however, this apparent gap is filled by the vicarious action of the planetary domiciles 
and exaltations that become effective, not only at the four cusps of the zodiac, but at 
the beginning of each of the twelve houses. 

(c) Domiciles and Exaltations (Dejections) 

In astrology, there are two competitive systems of attributing the signs of the 
zodiac to the seven planets, the "domicilia" and the i( exaltationes et dejectiones" . 
According to the first, each planet governs two signs, one during the day and one at 
night, with the exception of the Sun and the Moon, each of whom governs one sign 
only. According to the second, each planet has one sign in which he is "exalted", 
i.e. exerts his maximum action, and a sign diametrically opposite, in which he is 
"depressed" or "dejected", i.e. where his influence is least. As the co-existence of 
the two systems necessarily led to intolerable contradictions, it was agreed 35 that not 

36 Thus already in early astrological papyri (Mich. pap. No. 149, XVI, 2nd cent. A.D., and others, see Mich, Papyri, Vol. Ill, 
ed. J. Or. Winter, Ann Arbor, 1936, p. 116, and my review, in Isis, Vol. 27 (1937), pp. 337ff). Ptolemy, at the same time, 
seems to disregard the degrees. 



96 



The Mercury Horoscope of Marcantonio Michiel of Venice 



the whole sign, but only one single degree of it, and its close surrounding, should be 
counted as exaltation or dejection. The distribution is as follows : 





Domicilia 














Dejections 




Day 


Night 






Sun 


Leo 




Aries 


19° 


Libra 19° 


Moon . 




— ■ 


Cancer 


Taurus 


3 


Scorpio 3 


Mercury 




Virgo 


Gemini 


Virgo 


15 


Pisces 15 


Venus . 




Libra 


Taurus 


Pisces 


27 


Virgo 27 


Mars . 




Scorpio 


Aries 


Capricornus 


28 


Cancer 28 


Jupiter 




Sagittarius 


Pisces 


Cancer 


15 


Capricornus 15 


Saturn . 




Capricornus 


Aquarius 


Libra 


21 


Aries 21 


Ascending Node 


— 


— 


Gemini 




Scorpio 



A graphic illustration of the two systems is given in Figs. 6 (a) and (6). 

As mentioned before, the significance of the domicilia and exaltationes is not 
completely covered by the statement that the influence of the planets standing in 
or near to them becomes particularly powerful. For also alone, the signs and degrees 
in question are regarded as efficient, assuming the qualities of their lords. In 
horoscopes, therefore, they have to be taken into account and regarded as vicarious 
planets. 

Thus, examining the cusps of the twelve houses and comparing them with the 
above list, we find : 



House No. I 


Leo 


18° . 


II 


Virgo 


13 . 


III 


Libra 


10 . 


IV 


Scorpio 


6 . 


V 


Sagittarius 


11 . 


VI 


Capricornus 


14 . 


VII 


Aquarius 


18 . 


VIII 


Pisces 


13 . 


IX 


Aries 


10 . 


X 


Taurus 


6 . 


XI 


Gemini 


11 . 


XII 


Cancer 


14 . 



Sun's day domicile 

Mercury's day domicile, and Mercury exaltation — 2° 

Venus' day domicile 

Mars' day domicile, and Moon dejection + 3° 

Jupiter's day domicile, and Ascending Node dejection 

(no degree ascribed) 
Saturn's day domicile, and Jupiter dejection — 1° 
Saturn's night domicile 

Jupiter's night domicile, and Mercury dejection — 2° 
Mars' night domicile 

Venus' night domicile, and Moon exaltation + 3° 
Mercury's night domicile, and Ascending Node exaltation 

(no degree ascribed) 
Moon's night domicile, and Jupiter exaltation — 1° 



As may be seen, the distribution is such that the greater part of the day domiciles 
lie under the horizon, in the night half of the zodiac, and of the night domiciles, 
in the day half, so that only Leo and Aquarius, which are approximately bisected 
by the horizon, become really effective. By this choice, the astrologer has evidently 
succeeded in reducing the influence of the domiciles altogether in particular the evil 
effect of Mars at IMC, thereby giving full weight to the exaltations and dejections. 
Thus, from the point of view of vicarious planetary action, without respect to the 
actual positions of the planets themselves, of all the twelve houses only the following 
will be of astrological interest : 

I (Vita) : Sun's domicile, reinforced by rising. 
II (Lucrum) : Mercury's exaltation. 
VII (Uxor) : Saturn's domicile, extenuated by setting. 



Willy Hartner 



97 



X (Regnum) : Moon's exaltation. 
XI (Benefacta) : Ascending Node's exaltation. 
XII (Career) : Jupiter's exaltation. 

The astrological inferences to be drawn are the best possible ones : the royal lord 
of the planets, the Sun, governing the first house indicates good fortune and happiness 
in general. The cusp of the second house near the exaltation of Mercury, who is the 




A/is \ 

AC %\ 


or /§" \'\ 

wP^\A 

Z-^MOON Z"Y\ 




\ 27^C/ 

\ Av 



A: PLANETARY DOMICILIA 



B: PLANETARY EXALTATIONS 



Fig. 6. (a) Planetary domicilia; (b) planetary exaltations 



dominating planet of the horoscope, undoubtedly means wealth. Saturn's evil 
influence on the future marriage, in the seventh house, is reduced by the setting 
of the planet's domicile, Aquarius. The most important tenth house is determined 
by the exaltation of the benevolent Moon. The beneficent eleventh house is 
threatened by the exaltation of the evil Node (which, however, at the same time is 
heavily counterbalanced by the planetary double conjunction occurring in it). The 
danger of the maleficent twelfth house, finally, is averted by the exaltation of the 
good and friendly Jupiter. 

Now, passing over to the actual positions of the planets, whose influence, of course, 
prevails over that of their "deputies", we find that: 

IV (Genitor): Mars' evil influence extenuated (see above). 

VI (Valetudo) : determined by the propitious Moon. 

IX (Pietas) : determined by the ill-omened Saturn, whose effect, however, may 
be considered reduced by his proximity to dejection (distance 10°). 

XI (Benefacta) : overwhelming cumulation of luck 36 : the royal Sun and the 



36 It is true that planetary conjunctions, especially those with the Sun, are often regarded as unprosperous (but see below, 
Section 7). "The Sun burns or paralyzes the planet with which he stands in conjunction" (Bouche-Leclercq, op. cit., p. 245f ). 
At the same time, however, "'the planets in conjunction communicate to one another their good or bad qualities". In our case, 
Mercury being ambiguous (good with the good, and bad with the bad), the Sun being the symbol of creativeness and the source 
of the sensorial life — i.e. so-to-say beyond good and evil — and Jupiter, exclusively propitious, this double conjunction is hardly 
capable of an inauspicious interpretation. (As a parallel case, I refer to the laying of the first stone of Tycho Brahe'S observa- 
tory "Uraniborg" on the Isle of Hven, on 8th August, 1576, at sunrise, which date had been chosen expressly because it marked 
a conjunction of Jupiter with the Sun (ca. Leo 25°) near Regulus, both standing in trigonal aspect with Saturn (ca. Sagittarius 
24°), the Moon occupying ca. Aquarius 22°, i.e. being ca. 3° before her full. Though BRAKE, in Explicatio Partium majoris et 
praecipuae domus (Opera omnia, ed. I. C. L. Dreyer, Vol. V, Copenhagen, 1921, p. 143) mentions only the positions of the Sun, 
Jupiter, and the Moon ("Exoriente Sole una cum Jove juxta cor Leonis, Luna, occiduum cardinem in Aquario occupante"), 
he evidently had cast the whole horoscope so as to find a particularly auspicious day and hour. In any case, it leaves no room 

(See bottom page 98) 



98 The Mercury Horoscope of Marcantonio Michiel of Venice 

auspicious Jupiter in the House of Benefit, reinforced by Mercury, 
moving towards the exaltation of Jupiter (which is 13° distant, 
standing at the cusp of the next house). 

XII (Cancer) : the house of sorrow and trouble dominated by Venus — the most 
pleasant consolation that can ever be conceived ! 

(d) Aphetic Points and Aphets (Hyleg) 

The theory of genethlialogy, as established by Ptolemy and developed by the 
Arabs, pays special attention to the so-called aphetic points (tottol a<f>€TiKoL) occupied 
by a planetary aphet (d^drrjs) 37 . Boitche-Leclercq 38 calls this an "assimilation 
du Zodiaque a une roulette dans laquelle la vie des individus est lancee avec une 
force plus ou moins grande d'un certain point de depart (tottos a^en/co'?) et se 
trouve arretee, ou risque d'etre arretee, par des barrieres ou lieux destinateurs 
(toVoi avaiperiKol), sans pouvoir en aucun cas depasser un quart du cercle". From the 
position of an aphet in an aphetic point, most essential conclusions are drawn as to the 
life of the individual, particularly its duration. Of the twelve houses only five can 
become aphetic points, namely, in the following sequence: 

X (MC), I (Ascendant), XI (Bonus Genius), VII (Occasus), IX (called Deus). 
For births or conceptions occurring during the day, the gradation of aphets is : 

(1) The Sun, if standing in an aphetic point. 

(2) The Moon, if standing in an aphetic point. 
In case neither of the two fulfill the condition : 

(3) A planet standing in an aphetic point. If there are two or more competitors, 
special rules are to be observed to decide on the question of priority. 

(4) In the last resort, the house of the Ascendant is regarded as aphetic point. 
In that case, again, special prescriptions exist for deriving the aphet from the Ascen- 
dant (he may be identified with the lord of the domicile, or of the exaltation, finis, or 
the like). 

For births occurring at night, the Sun and the Moon, in the above gradation, are 
interchanged. 

In the present case, evidently, the Sun occupying the locus boni genii (House 
No. XI) is to be regarded as aphet of the horoscope. 

(e) The Wheel of Fortune (kXtjpos tvx^s, Fortuna) 39 

It seems astonishing that this chimera of early astrological (allegedly Egyptian) fancy 
does not occur on the horoscope. The "Wheel of Fortune" is an imaginary point to 
which auspicious planetary qualities are ascribed ; it is commonly represented by the 
figure of a cross in a circle ("wheel"), 0. It holds the degree of the ecliptic whose 
angular distance from the Moon (counting with increasing longitudes) is the same as 



(Continuation of footnote 36 from page 97) 

for doubt that the sixteenth century astrologers regarded conjunctions of the beneficent planets with the Sun as strictly 
favourable.) In the case of Marcantonio's horoscope, the conjunction takes place in the locus boni genii (XI), the auspiciousness 
of which is increased by the Sun (who is the aphet of the horoscope, see below, Section 5 (d)). But it should not be forgotten 
that most of the rules of astrology are vague enough to allow of different, often strictly contradictory, interpretations. Two 
astrologers dealing with the same case will seldom, if ever, arrive at identical results. Therefore, it is hardly necessary to state 
that my inferences from the given facts have no claim to absolute certainty. I may have overlooked details to which the 
Renaissance astrologer ascribed particular importance, and I may have unduly stressed others which, according to the taste of 
the time, were considered negligible. 

37 Renaissance astrologers use the term hyleg or alhyleg, derived from the Persian hailaj, "the one who lets loose", a literal 
translation of the Greek dtperrjr (Persian — HIL — , = Greek apiy/xt). The stress laid on the hyleg in early European] astrology 
is obviously due to Islamic influence. 

38 Op. cit., p. 411; see also al-Battani, Opus Astronomicum (ed. Nallino), Vol. I, p. 313f. 

39 Cf. Botjche-Leclercq, op. cit., pp. 288-310. 



Willy Habtnbb 



99 



that of the Ascendant from the Sun (A — Aj = A ASC — X Q ). The house occupied by 
it becomes prosperous even if it be that of death or evil spirits. Et si fuerit in loco 
malifortuna verte sentenciam et die loco mali bonum, we read in the Flores Albumasaris i0 . 
In the present horoscope, the Wheel of Fortune would be in 336° = 6° Pisces 
(below horizon), in the night domicile of Jupiter, in the finis governed by Venus, and 
in the fades of Saturn. As it is simultaneously in the House of Marriage (VII, 
Occasus), additional information on the future marriage of Marcantonio's child can 
be drawn from it. The prognostication is evidently propitious, because the evil 
influence of Saturn is heavily counterbalanced by the accumulation of auspicious 
elements. 



6. Secondary Elements of Prognostication: Limits (Fines), Decans, 

and trines (trigona) 

Even the statements made hitherto do not exhaust the possibilities of astrological 
prognostication 41 . The "secondary elements" may sometimes be just as important 
as those treated above, and it would, therefore, be a serious omission if we did not 
take some of them into consideration. 

(a) Limits (Lat. fines) 

These result from a subdivision of each of the twelve signs of the zodiac into five 
unequal parts, each of which is governed by one of the five planets (taking no account 
of the Sun and the Moon). There are three main systems competing with one another, 
the "Egyptian", the "Chaldean", and that worked out by Ptolemy. Only the first 
of them, however, seems to have been used in practical astrology of the Arabic 
Middle Ages and the Renaissance 42 . According to these Egyptian rules the first finis 
of Aries, comprising 6°, is governed by, or stands as a substitute for, Jupiter; the 
second, comprising 6°, belongs to Venus; the third (8°) to Mercury; the fourth (5°) 
to Mars; the fifth (5°) to Saturn, etc., as is seen from the following table. 





I 


II 


III 


IV 


V 


Aries 


Jupiter 


6° 


Venus 6° 


Mercury 8° 


Mars 5° 


Saturn 5° 


Taurus . 


Venus 


8 


Mercury 6 


Jupiter 8 


Saturn 5 


Mars 3 


Gemini . 


Mercury 


6 


Jupiter 6 


Venus 5 


Mars 7 


Saturn 6 


Cancer . 


Mars 


7 


Venus 6 


Mercury 6 


Jupiter 7 


Saturn 4 


Leo 


Jupiter 


6 


Venus 5 


Saturn 7 


Mercury 6 


Mars 6 


Virgo 


Mercury 


7 


Venus 10 


Jupiter 4 


Mars 7 


Saturn 2 


Libra 


Saturn 


6 


Mercury 8 


Jupiter 7 


Venus 7 


Mars 2 


Scorpio . 


Mars 


7 


Venus 4 


Mercury 8 


Jupiter 5 


Saturn 6 


Sagittarius 


Jupiter 


12 


Venus 5 


Mercury 4 


Saturn 5 


Mars 4 


Capricornus . 


Mercury 


7 


Jupiter 7 


Venus 8 


Saturn 4 


Mars 4 


Aquarius 


Mercury 


7 


Venus 6 


Jupiter 7 


Mars 5 


Saturn 5 


Pisces . 


Venus 


12 


Jupiter 4 


Mercury 3 


Mar3 9 


Saturn 2 



(b) The Thirty-six "Faces'' or "Decans" (Lat. "fades") 

This is another subdivision of each of the twelve signs into three equal parts 
("decans"), which can be traced back to early Egyptian prototypes, being the main 



40 Fol. 7v-9r (passim), Augsburg, 1488 (facsimile reprint by Deutscher Verein f iir Buchwesen und Schrifttum, Leipzig, 1928). 

41 "Les d6bris de toutes les fantaisies qui n'avaient pas trouve place dans la repartition des oIkoi ("houses") et des vtycifixra 
("exaltations") ont servi a fabriquer le systeme des trigones planetaires. On s'achemine peu a peu vers l'incompreliensible, qui 
atteint sa pleine floraison dans le systeme des opia ("limits")", says Botjche-Leclercq, op. cit., p. 198f. 

48 Thus, for example, on astrolabes; cf. W. Hartner, "The Principle and Use of the Astrolabe", in A Survey of Persian Art 
(ed. A. U. Pope), Vol. Ill, Oxford, 1939, Ch. 57, p. 2548ff. 



100 



The Mercury Horoscope of Marcantonio Michiel of Venice 



contribution of pre -Hellenistic Egypt to mediaeval astrology 43 . Again, each of the 
decans is governed by one of the planets — this time including the Sun and the 
Moon — as illustrated by the following table. 





I 


II 


III 




I 


II 


III 


Aries 


Mars 


Sun 


Venus 


Libra 


Moon 


Saturn 


Jupiter 


Taurus 


Mercury 


Moon 


Saturn 


Scorpio 


Mars 


Sun 


Venus 


Gemini 


Jupiter 


Mars 


Sun 


Sagittarius 


Mercury 


Moon 


Saturn 


Cancer 


Venus 


Mercury 


Moon 


Capricornus 


Jupiter 


Mars 


Sun 


Leo . 


Saturn 


Jupiter 


Mars 


Aquarius . 


Venus 


Mercury 


Moon 


Virgo 


Sun 


Venus 


Mercury 


Pisces 


Saturn 


Jupiter 


Mars 



The practical application of the two systems to our case is shown in the following 
table, where the longitudes of the cusps of the twelve houses are listed in the left-hand 
column, the corresponding fines with their lords in the middle, and the decans with 
their lords in the right-hand column. 





Finis 


Tacies 


House No. I (Ascendant) 


. Leo 


18° 


IV, Mercury 


II, Jupiter 


II 


. Virgo 


13 


II, Venus 


II, Venus 


III 


. Libra 


10 


II, Mercury 


I, Moon 


IV (IMG) . 


. Scorpio 


6 


I, Mars 


I, Mars 


V . 


Sagittarius 


11 


I, Jupiter 


II, Moon 


VI . 


Capricornus 


14 


III, Venus 


II, Mars 


VII (Occasus) 


Aquarius 


18 


III, Jupiter 


II, Mercury 


VIII . 


Pisces 


13 


II, Jupiter 


II, Jupiter 


IX . 


Aries 


10 


II, Venus 


II, Sun 


X (MG) . 


Taurus 


6 


I, Venus 


I, Mercury 


XI . 


Gemini 


11 


II, Jupiter 


II, Mars 


XII . 


Cancer 


14 


III, Mercury 


II, Mercury 



Any astrologer who contrived to select a similarly auspicious Ascendant would, 
indeed, be worthy of praise. Of the two malevolent planets, Saturn is completely 
banished from the fines as well as the fades, whereas Mars occurs only once in the 
former, and three times in the latter. All of the rest are governed either by Mercury 
or by the two benevolent planets (Jupiter and Venus), the Sun occurring only once, as 
lord of the second decan of Aries, at the cusp of House No. IX (Pietas). 

In particular, the conjunction of the Sun with Mercury and Jupiter is "repeated" 
by the Ascendant being simultaneously in the domicilium of the Sun, in the finis of 
Mercury, and in the fades of Jupiter. Leo 18° marks exactly the beginning of the 
fourth finis (extending from 18°-24°); hence the first 25 m approximately (corres- 
ponding to 6° in longitude, or a little more in AR) subsequent to the moment for 
which the horoscope was set, will be dominated by the lord of the horoscope, Mercury. 
Again, Mercury, together with Jupiter, appears at the Descendant, and, with Venus, 
at the MC. Finally, he concludes the "circulus geniturae" as the double lord of the 
finis and the fades of the cusp of the twelfth house, whereby the evil influence of this 
locus mali genii, already extenuated by Jupiter's exaltation (see above, Section 5 (c)), 
is minimized. 



1936. 



Cf. W. Gundel, "Dekane und Dekansternbilder" (Studien der BiUiothek Warburg, Vol. XIX), Gliickstadt and Hamburg, 



Willy Hartner 101 

Only the danger threatening on the part of Mars cannot be completely disregarded. 
The fact that he is Lord of the finis and the fades standing at the cusp of the fourth 
house {IMC, Genitor), moreover that this house coincides with Scorpio, his domicilium, 
and that it is actually occupied by the planet himself, necessarily implies that the 
father of the child to be born will have to face martial hardships. The other threat, 
which arises from Mars' being connected with the decisive eleventh house, where the 
great conjunction takes place, is counterbalanced to a great extent by the auspicious 
conjunction itself, but is probably not completely negligible because of the Ascending 
Node's exaltation. Nevertheless, as a whole, the astrological situation seems extra- 
ordinarily, not to say uniquely, auspicious. 

The reader who has followed my analysis thus far is entitled to an explanation 
of the really amazing fact that the astrologer, in selecting the day and the hour, con- 
trived to make a number of heterogeneous circumstances tally so perfectly that the 
propitious symptoms of the horoscope by far prevail over the unpropitious ones. One 
part of the answer is given by referring to the fact that, above all, the auspicious 
nature of the day offers itself almost automatically, for not only the double conjunc- 
tion but also all the other actual planetary aspects (see below, under (c), and Section 7) 
appear to be as favourable as they can be. Hence particular astrological skill is indeed 
required only in selecting the Ascendant in such a way that the propitious sections 
or points of the zodiac coincide with the cusps of the twelve houses, while the 
unpropitious ones are banished from the critical points or at least counterbalanced 
by one or the other accessory element. Obviously, it seems a laborious, not to say 
"sisyphean", work to find the most appropriate solution by way of calculation, 
because for each ascendant we obtain a different distribution of houses and, con- 
sequently, also different lords of fines and fades, as well as different effects of the 
domidles and exaltations. It is true that the distribution of the planets among the 
fines (cf. the above table) is apt to facilitate the choice, because the three first fines 
of each sign (i.e. approximately the first 20° of each) are prevalently governed by 
auspicious planets (exceptions being only the following four out of thirty-six: 
Cancer I, Mars; Libra I, Saturn; Scorpio I, Mars; and Leo III, Saturn), whereas 
the ill-famed planets appear in the two last fines. But as there is no similar regularity 
in the fades and because, on the other hand, the deviation of the houses from the 
average extent of 30° can be considerable, the astrologer's task still seems difficult 
enough. 

The explanation, no doubt, is that a rough approximation was obtained with the 
aid of an astrolabe. This most ingenious instrument of the Middle Ages enabled the 
astrologer to solve these or similar problems simply by way of systematic trial. The 
work of days or weeks could thus be done in less than an hour. A first approximation 
being found, the astrologer would resort to the manifold tables at his disposal 44 in 
order to establish the definitive value of the Ascendant. Thus actual calculation 
could be reduced to a minimum or even disposed of completely. 

(c) Trines (Lat. trigona or triquetra) 

They consist of four groups of three signs of the zodiac lying 120° apart; each trine 
is attributed to one of the four elements, and governed by one lord of the day, one 
of the night, and one companion. 

44 Tables of right ascensions for each degree of the ecliptic, of oblique ascensions in the seven climates and for the latitudes 
of important cities, etc.; cf. al-Battani, op. cit., Vol. II, pp. 61-71, also Die astronomischen Tafeln des Muhammed ibn Musd 
al-Khwarizml (ed. H. Sitter), Copenhagen, 1914, Tables 79-90 (pp. 194-205). 



102 



The Mercury Horoscope of Marcantonio Michiel of Venice 





Element 


Lord of 


Companion 


Signs 


Day 


Night 


Trine I Aries, Leo, Sagittarius 

II Taurus, Virgo, Capricornus 

III Gemini, Libra, Aquarius . 

IV Cancer, Scorpio, Pisces 


Fire 
Earth 
Air 
Water 


Sun 
Venus 
Saturn 
Venus 


Jupiter 
Moon 
Mercury 
Mars 


Saturn 
Mars 
Jupiter 
Moon 



In the present case, we find : 

Trine I: not occupied by planets (the Ascending Node in Sagittarius does not 
count). 

Trine II : (1) Saturn in Taurus. This is of no interest because Saturn is neither lord 
nor companion. (2) Moon in Capricorn. Of decisive importance, 
because the auspicious Moon which governs the night stands in the 
night part of the trine (Capricorn below the horizon). Highly propitious 
symptom for prognostication. 

Trine III : not occupied by planets. 

Trine IV: (1) Sun, Jupiter, Venus, and Mercury, in Cancer (above horizon). 
Very propitious because Venus as lord of the day stands in the day part 
of the trine, supported by three other benevolent planets. (2) Mars as 
lord of the night in the night part, is powerful but not obnoxious because 
he is neutralized by the propitious cumulation of planets in Cancer. 



7. Actual Planetary Aspects 

It is due to practical considerations only and not to intrinsic reasons, that I am 
treating this question only after having dealt with the influence of the various 
fictitious divisions and subdivisions of the zodiac {domiciles, exaltations, houses, fines, 
fades, trines). Evidently, the alleged effect produced by the relative positions 
(angular distances) of the planets ranks above that of vicarious points and sections, 
and will, therefore, have to be considered first by the astrologer. 

The rules are simple. An angular distance (difference in longitude) of 180° is called 
opposition, as in astronomy; 120° make the trigonal aspect; 90° the quadrature; 
and 60° the sextile aspect. Oppositions and quadratures are considered unfavourable, 
especially when ill-omened planets are involved, and the latter more so than the 
former, while the trigonal and sextile aspects are regarded as propitious phenomena. 

The conjunction, in classical astrology, was not counted among the aspects, strictly 
speaking, but 45 starting with the early Islamic period it played an important, even a 
decisive role. The famous Arabic astrologer Albumasar (Ja'far b. Muhammad b. 
'Umar Abu Ma'shar al-Balkhi), who died at the age of 100, in 886 a.d., was one of 
the first to compile a bulky book specially devoted to conjunctions, which was copied 
and commented on innumerable times by later astrologers. In the translation of 
Johannes Hispalensis (fl. about 1135-1153), this work of Albumasar's was among 
the first to be published in print 46 . A first edition appeared simultaneously in Augs- 



45 Cf. footnote 36, page 97. 

46 Albumasar de magnis conjunctionibus et annorum revolutionibus ac eorum profectionibus octo continens tractatus. The Flores 
Albumasaris ("Tractatus Albumasaris florum astrologie"), mentioned in Section 5 (e) which were printed by Ratdolt in 1488 
are possibly an extract from the great work. 



Willy Hartner 



103 



burg (printing office of Erhard Ratdolt) and in Venice ; a second was published in 
Venice in 1515. 

The fact that the City of Venice, in the course of one generation, witnessed the 
publication of two editions of this work, obviously bears upon the present problem. 
Or perhaps it is only chance that twelve years after the publication of the second 
edition, the patrician Marcantonio of Venice ordered a horoscope to be worked out 
in which a conjunction of two planets with the Sun holds the central place 47 ? It would 
seem a promising task to examine Albumasar's great work from this point of view to 
find out what prognostication, according to him, can be inferred from Marcantonio's 
double conjunction. This new problem, however, will not be treated here ; it may be 
made the subject of another study, insha'llah. 

In the present case, the mutual angular distances between the planets are as 
follows. 



Sun-Moon (198°) 


162° 


Moon-Venus (186°) 174° 


Venus-Msirs 117° 


—Mercury 





-Mars (303°) 57 


-Saturn (275°) 85 


-Venus 


24 


-Saturn 101 


—Ascending Node 148 


-Mars 


141 


-Ascending Node (334°) 26 




-Jupiter 







Mars-Saturn (202°) 158° 


-Saturn (299°) 


61 


Mercury and Jupiter (see Sun) 


—Ascending Node 31 


-Ascending Node 


172 




*Sta«rn-Ascending Node (233°) 127 



Apart from the conjunction, only the following are of interest from the point of 
view of aspects : 

Sun (with Jupiter and Mercury )-Saturn : sextile aspect + 1°. Strictly favourable, 
because binding the dangerous Saturn. 

Sun-Ascending Node : 8° past opposition. The danger threatening from opposition 
has gone. 

Moon- Venus : 6° before opposition. The beneficent effect of the two planets is not 
yet blocked by the opposition occurring ca. ll n later. 

Moon — Mars: 3° before sextile aspect. The Moon will reach the favourable aspect 
within 6 or 7 hours. The evil influence of Mars will, therefore, be banished until the 
afternoon of the subsequent day 48 . 

Moon-Saturn: 11° before quadrature. The Moon has just passed the limit of 13°, 
where the evil effect of her quadrature with Saturn may become perceptible. But the 
danger is still minimal. 

Venus-Mars : 3° past trigonal aspect 49 . The prosperous effect of the aspect still 
lasting. 

Venus-Saturn : 5° before quadrature. The evil effect of the aspect not yet percep- 
tible. 

In other words, this last of the main elements of prognostication (primary and 
secondary) is just as auspicious as all the preceding ones. It would probably be hard 



47 But the doctrine of conjunction played a big part all through the fifteenth century. (See also A. Warburg's studies on 
Martin Luther in "Heidnisch-antike Weissagung in Wort und Bild zu Luther's Zeiten" in Sitzungsberichte d. Heidelberger 
Akad. d. Wiss., Philol.-histor. Kl., 26. Abh., 1919, e.g. pp. 12 and 80; furthermore G. Bing (Editor), A. Warburg. Gesammelte 
Schriften, Band II, Teubner, Leipzig, 1932. See also F. Boll and C. Bezold, Sternglaube und Sterndeutung ; die Geschichte und 
das Wesen der Astrologie, 4th Edition; e.g. pp. 34 and 117; Teubner, Leipzig, 1931.) 

48 According to Bouche-Leclercq, op. cit., p. 245, the effect of conjunctions is perceptible in an area extending 3° to both 
sides of the point of contact as far as the Sun and the five planets (except the Moon) are concerned. It seems reasonable to 
apply the same rule to the other aspects. In the case of the Moon, the sphere of influence extends to ± 13° from the critical 
points (aspects), corresponding to the distance travelled in the course of one day. 

4 * Venus in direct motion, between superior conjunction and maximum eastern elongation. 



104 



The Mercury Horoscope of Marcantonio Michiel of Venice 



to find a second day and hour in Marcantonio's life that marks a similarly propitious 
constellation. It would seem perfectly reasonable, therefore, that our Venetian 
patrician considered this horoscope worthy of being engraved on the marble statuette 
representing his astrological patron and protector. 

8. The Probable Date of Marcantonio's Marriage 
Once our curiosity has been roused, we find it difficult to stop at this point. For does 
it not seem probable, nay certain, that Marcantonio would also allow himself to be 
guided by astrological considerations in fixing the date of his marriage ? 

All we know is that the marriage was celebrated in February, 1527 50 . Considering 
the fact that the horoscope of 15th June was dominated by the conjunction of 
Mercury with the Sun and Jupiter, we may duly expect another striking phase of 
Mercury — possibly also a conjunction — to play an analogous part in the marriage 
horoscope. Of course, a double conjunction would be highly unlikely (indeed, there 
is none during the whole month). The only astrological phenomenon to be con- 
sidered, therefore, would be a conjunction of Mercury either with the Sun or with 
Venus, which is even more auspicious than Jupiter. In fact, both occurred in 
February, 1527, the former about 7th February, the latter, 11th February. In 
checking the astrological situation of the two days, it has been found by means of 
approximate calculations that only the second one is capable of a favourable inter- 
pretation. 





7th February, 1527 (noon) 


11th February, 1527 (noon) 




(Conjunction Mercury-Sun) 


(Conjunction Mercury-Venus) 


Sun 


328° (Aquarius 


28°)* 


332° (Pisces 


2°) 


Moon . 






45 (Taurus 


15 ) 


98 (Cancer 


8 ) 


Mercury 






328 (Aquarius 


28 )* 


325 (Aquarius 


25 ) 


Venus 






320 (Aquarius 


20 ) 


325 (Aquarius 


25 ) 


Mars . 






236 (Scorpio 


26 ) 


238 (Scorpio 


28 ) 


Jupiter 






72 (Gemini 


12 ) 


73 (Gemini 


13 ) 


Saturn 






13 (Aries 


13 ) 


13 (Aries 


13 ) 


Ascending Node 




271 (Capricornus 


1 ) 


271 (Capricornus 


1 ) 



* Exact values (mean noon Venice): Sun 327° 52', Mercury 328° 7'. 

The difference between the two consists mainly in the changed position of the 
Moon. On 7th February, her good influence is blocked by the approaching opposition 
with Mars, which will occur within 24 h , while on 1 1th February, she not only occupies 
her domicilium (Cancer), but also stands close to the trigonal aspect with the Sun. 
It is true that the danger of her quadrature with Saturn is imminent, but it could be 
averted by choosing the later part of the preceding night for the marriage ceremony, 
when the Moon entering Cancer is still 13° distant from the critical point. Equally 
unprosperous, in both cases, is the proximity to quadrature with Mars, on 7th 
February, of the Sun and Mercury, on 11th February, of Mercury and Venus. But 
as the figures given above are only rough approximations, and as undoubtedly they 
do not necessarily agree with those derived from contemporary astronomical tables, 
it would be unjustified to draw further conclusions. 

The most powerful argument for 1 1th February as the probable wedding day — or 
rather a day near to 11th February on which, according to the unknown tables used, 



See Section 4, page ; 



Willy Hartner 105 

the conjunction of Mercury and Venus took place — is that no believer in astrology 
who has chosen Mercury as his patron, will fail to celebrate his marriage on the day on 
which his planet-god celebrates his conjunction with the goddess of love, Venus 51 . 
And should there be evil aspects threatening to destroy the harmony of the day, a 
skilled astrologer will always find ways to explain them away or twist them so as to 
make them form a friendly picture. For, here as elsewhere, it would appear that the 
real aim of man is not to learn the truth but to have confirmed what he would like 
to be the truth. Mundus vult decipi 52 . 

II. ASTRONOMICAL PART 

9. The Ptolemaic Theory of Planetary Motion 

(a) Venus, Mars, Jupiter, and Saturn 

It is not without pride that Ptolemy (ca. 140 a.d.), in his Mathematical Syntaxis 53 
(which we are accustomed to call by its corrupted Arabic name, Almagest) claims to 
have been the first to work out a complete mathematical theory of the planetary 
motions. In the second chapter of the 9th book 54 , he pays tribute to his predecessor, 
Hipparchus (ca. 150 B.C.), "this great lover of Truth", who, occupying himself 
very thoroughly with the theory of the Sun and Moon, had proved that the orbits of 
these two luminaries can be represented on the basis of the (Aristotelian) postulate of 
uniform circular motion in the eccentric deferent as well as in the epicycle. For the 
planets, however, Hipparchus had only shown that this postulate does not suffice 
to explain their complicated motions, as determined by observation. 

For this reason, Ptolemy introduces the new hypothesis that the centre of the 
epicycle moves on the eccentric with varying velocity, in such a way that only when 
seen from the punctum aequans — i.e. the point on the line of apsides whose distance 
from the Earth is the double of the linear eccentricity — the motion in the eccentric 
appears to be uniform. The characteristic features of the Ptolemaic hypothesis as 
applied to the planets Venus, Mars, Jupiter, and Saturn, are illustrated by Fig. 7, 
where 

E = Earth, centre of ecliptic, AXYA-^D, 

F = centre of eccentric deferent, GHK, 

E' = punctum aequans, 

EF = FE' = linear eccentricity 55 , 

AA X = line of apsides, 

O = apogee of deferent, 

X = perigee of deferent, 



51 See "Concluding Notes" at end of study. 

63 The period under discussion here is only some sixty years later than that of the Chigi-Horoscope of 1466, on the ceiling of 
the Sala di Galatea of the Farnesina in Rome, the investigation of which was carried out in 1934 by Fritz Saxl and Arthur 
Beer (Reale Accademia d'ltalia, Collezione "La Farnesina", Home, 1934, XII): Part I, by F. Saxl, "La fede astrologica di 
Agostino Chigi"; Interpretazione dei dipinti di Baldassare Peruzzi nella Sala di Galatea della Farnesina (57 pp.). Part II, by 
A. Beer, "II significato astronomico e la data dei dipinti della volta della Sala di Galatea" (7 pp.). It is interesting to compare 
the historical and personal aspects as they appear from this interpretation of Chigi'S horoscope with those set out in the 
present paper. 

53 Greek text: CI. Ptolemaei Syntaxis Matkematica, ed. J. L. Heiberg, Part I (Books 1-VI), Leipzig, 1898; Part II (Books 
VII-XIII), 1903. German translation: Des CI. Ptolemitus Handbuch der Astronomie, aus dem Oriechischen ubersetzt von K. 
Manitius, Vol. I, Leipzig, 1912; Vol. II, 1913. Quotations (Almagest) are made from the German translation. 

54 Almagest, Vol. II, p. 96rT. 

65 Expressed in parts (60th) of the radius FO of the deferent (all other linear quantities, such as the radius of the epicycle, 
etc., are expressed in the same way). 



106 



The Mercury Horoscope of Marcantonio Michiel of Venice 



H — centre of epicycle, gTk, 
g = mean apogee of epicycle, 
Jc = mean perigee of epicycle, 
T = planet, 




l_AE'Z 
l_AEX 
Z_XHZ 



LgKT 

l_XEY 



Fig. 7. Venus and superior planets 

longitude of apogee O of deferent, 

(J) = mean anomaly of centre H of epicycle, 

v = apparent 56 anomaly of centre H of epicycle, 

Z^EHE' = "zodiacal inequality" (Ptolemy: -q E,co8lclk7) dVco/xaAi'a, in 
Arabic : Ta'dll al-hdssa wa!l-markaz, whence mediaeval Lat. aequatio 
anomaliae et centri, and in modern usage, "equation of the centre"), 

r = (mean) anomaly of planet T in the epicycle, 

aequatio anomaliae {avoy^aXias Trpoa&a^aipeais, "prosthaphairesis"). 



The apogee of the deferent is supposed to have a uniform direct motion round E, 
its amount being the same as that of precession (1° in 100 a ) ; in other words, the line 
of apsides is fixed relative to the fixed stars. The revolution of H on the deferent, as 



66 Apparent always to be understood as "appearing to the eye", i.e. as actually observed from the Earth. 



Willy Hartner. 107 

mentioned before, is uniform in regard to the punctum aequans, E' , not the centre of 
the deferent, F. Finally, the planet itself revolves uniformly — and equally counter- 
clockwise, as all other points — in the periphery of the epicycle 57 . Thus the tropical 
revolution of the planet is represented by the motion of H in the deferent (called 
IatJkos, [motion in] longitude) 58 , whereas the synodic is taken care of by that of the 
planet in the epicycle (avu>pa\la, anomaly). In Aim. IX. 4 59 , the longitudes and 
anomalies are tabulated for periods of 18 a , single years, months, days, and hours, of 
the Egyptian calendar (l a = 12 months of 30 d + 5 epagomenal days, without 
intercalation), the epoch being Thoth 1 of the 1st year of Nabonassar, noon 
= 26th February, 747 B.C. The reduction of mean to apparent longitudes is effected 
with the aid of special tables, one for each planet (Aim. XI. 12 60 ), arranged in such a 
way that the aequatio centri as well as the aequatio anomaliae can be obtained from 
them by double interpolation. 

Ptolemy's theory of latitudes is developed in the six first chapters of Aim. XIII 61 . 
It will be disregarded here as it does not bear directly on our subject. 

(6) Mercury 

The mathematical hypothesis demonstrated above yields fairly good results in the 
case of the four planets, Venus, Mars, Jupiter, and Saturn. It is, however, not 
sufficient to represent the motion of Mercury, which, on account of the considerable 
eccentricity of its orbit, was of crucial importance for ancient and mediaeval astro- 
nomers. They were particularly puzzled by the circumstance that, in the course of 
one revolution, the centre of the epicycle passes twice through a perigee point, but 
only once through an apogee (see Section 10). 

Being aware of the impossibility of obtaining satisfactory results by operating 
with a deferent having a constant eccentricity, Ptolemy devises the following 
modification of his simple hypothesis. In Fig. 8 the line AA 1} corresponding to the 
line of apsides in the preceding figure 62 , follows the precession of the equinoxes, i.e. is 
considered immovable relative to the fixed stars. 

On it, E marks the Earth, and E' the punctum aequans (being the centre of the 
circulus aequans, or "equant", a), from which the revolution of the centre of the 
epicycle appears uniform. The centre of the movable deferent lies on the periphery 
of a small circle MM'E' the centre of which, F, has the same distance from E' as 
E' from E. 

Starting from a certain moment, t, when the centre H of the epicycle coincides 
with the apogee G of the deferent in its initial position on the apse-line (dotted 
circle, d, with centre M), the centre of the deferent, in the course of one tropical year, 
revolves with uniform retrograde motion on the small circle around F, while the 
radius vector E'H simultaneously carries around, with uniform direct motion, the 
centre H of the epicycle. At the moment t' , the centre of the deferent (circle d') 
being in M' , and its apogee in G', the centre of the epicycle will occupy H', so that 
/J}FG' = /_GE'H' = (f>. After one half-year, the centre of the deferent coincides 



t, IX.6 (Vol. II, p. 123). 
In the case of Venus, as with Mercury, the period of revolution of H is the tropical year because the longitude of H is 
always equal to the Sun's mean longitude ; see below, Section 9 (b), 

59 Almagest, Vol. II, pp. 104-118. 

60 Almagest, Vol. II, pp. 261-265. 

61 Almagest, Vol. II, pp. 325-380. 

62 Contrary to the other planets, the point A x is not the perigee point, whence the term "line of apsides" is not applicable 
to this case. Only for the sake of brevity, the line AA X will henceforth be called "apse-line." 



108 



The Mercury Horoscope of Marcantonio Michiel of Venice 



with the punctum aequans, E', and the deferent itself, with the "equant" a, and thus 
the centre H" of the epicycle and the movable apogee, G" , will again meet on the 
apse-line, but this time between E and A x . 




Fig. 8. Mercury 



As in the preceding case, the planet itself revolves uniformly in the epicycle. The 
period of revolution of the centre of the epicycle is the tropical year, as also in the 
case of Venus, and thus the centre of the epicycle always has the same mean longitude 
as the Sun 63 . This, however, does not necessarily imply that the Sun itself is regarded 
as the centre of revolution. On the contrary, Ptolemy and his followers obviously 
take no notice of the (so-called) "Egyptian" hypothesis as propagated by 
Heracleides, according to which Mercury and Venus are believed actually to 
revolve around the Sun. They disregard or reject it with good reason indeed, because 
evidently this naively devised hypothesis becomes improbable and useless as soon as 
the true Sun is replaced by the mathematical point representing the Sun's mean 
longitude. 

The equation of the centre (/_E'H'E) and that of anomaly (/_H'ET') are com- 
puted and tabulated in the same way as in the case of the other planets. From his 
observations, Ptolemy derives the following values for the constants by which the 



Almagest, IX.3 (Vol. II, p. 102, ] . 17-19), see footnote, 58, page 107. 



Willy Habtner 109 

orbit of Mercury is determined, taking again no account of its inclination as not 
relevant to our subject) : 

Linear eccentricity e = 3 P 0' 64 = 0-05.R (R = 1). 

Radius of epicycle P = 22 p 30' = 0-375R. 

Longitude of apogee A = Libra 1° 10' = 181° 10'. 

Mean longitude of centre of epicycle L = Pisces 0° 45' = 330° 45' (= mean 
longitude of Sun). 

Anomaly (counted from mean apogee of epicycle) r — 21° 55'. 

(Epoch of A Q , L , t = 1st year of Nabonassar, Thoth I, noon = 26th February, 
747 B.C. 65 .) 

10. Excursus: The Curve Described by the Centre of the Epicycle. 
Ibn al-Samh and Azarquiel, The Libros del Saber, and Peurbach 

The Almagest contains no statement concerning the nature of the curve which the 
centre of the epicycle actually describes according to Ptolemy's theory of Mercury. 
To the best of my knowledge, the first European author to speak about it explicitly 
is Georg Peurbach (1423-61). In his Theoricae novae planetarum 66 , he refers to it 
with the words: "Sexto ex dictis apparet manifeste centrum epicycli Mercurii 
propter motus supra dictos non ut in aliis planetis fit: circumferentiam deferentis 
circularem sed potius figurae habentis similitudinem plana ovali periferiam describere". 



This statement is perfectly true, for under the given conditions I "linear" eccen- 

R\ ^ 

tricity e = 3P = — - 1 , as will be seen, the curve is practically identical with an 

ellipse having F as its centre, and a — R + e and b — R — e as its axes (eccentricity 

according to modern definition e = — = 0-4259180. . . ). Theoretically, however, 

it is an algebraic curve represented by the polar equation 



r = e(cos cf> + cos 2<f>) + Vi? 2 - £ 2 (sin + sin 2<£) 2 , .... (1) 

where the punctum aequans, E' , marks the origin, and E'A, the axis, of a system 
of polar co-ordinates, r and (see Fig. 9). 

My assertion that, for the Ptolemaic ratio of its two parameters, R = 20s, this 
curve becomes practically identical with an ellipse, is not a matter of course and has 
to be proved. Since a complete analysis of its mathematical properties is beyond our 
scope, I limit myself to the following observations. 

For small values of the ratio — , the curve is pear-shaped, as shown in Fig. 9 67 . 



£ 

T _,,. . . n . . du , . n , „ du 

Letting u = sin <f> + sin 2j>, and — = cos + 2 cos 2<f>, we find, from — = for 

dcp dcp 

4> M = dz 53°625, that u reaches its maximum, u M = + 1-76017, for this value of </> 68 . 

" IP = 60' = — of radius of deferent, R = 60P, see footnote 55, page 105. 

65 Nallino, in al-Battani, Opus Astronomicum, Vol. I, p. 241, erroneously assumes the year 137 a.d. to be the epoch to 
which Ptolemy reduces his observations, though it is expressly stated, Almagest, IX.ll (Vol. II, p. 155 et passim), that the data 
are consistently referred to the beginning of the era of Nabonassar. Therefore the differences between Ptolemy's values for the 
apogees of Saturn, Jupiter, and Mars, and, on the other hand, those computed from Leverriek'S elements are about 10° 
smaller than indicated by Nallino (2°-5°, instead of 12°-15°). 

66 Editio princeps, printed by Regiomontanus, Nuremberg, 1472, fol. 21f. 

67 As the curve evidently has the line AE as its axis of symmetry, only the left half of it is drawn, for various ratios — . 

68 u attains its minimum u m = — 0-3690 near <t> = ± 14795. Because \u m \ < \u M \, this value is of no interest to us. 



110 



The Mercury Horoscope of Marcantonio Michiel of Venice 



Hence the curve will become undefined in the neighbourhood of cf> M , for R < e . u M . 
The equation 

dr £ 2 {sin </>(cos <j> + cos 26) + sin 3</> + sin 4=6} 

— = — «(sin (f> + 2 sin 26) = = 

d6 v r i r/ ^^2 _ g2(gin ^ + gin 2 ^ a 

....(2) 

has the trivial solutions = 0° and 6 = 180°, indicating that r has two maxima 69 , 
viz. r = R + 2e for 6 = 0° and r = R for 6 = 180°. 







A 






M^\P 




7/ \ 








F / J 


T~~~~~---+^^^^^^^\ 






E' 




E 






A, 



Fig. 9. The Ptolemaic curve for small 



R 



EE' = E'F = e 

M J. = PH = R 

LAFP = /_AE'H = 



LAEH. = v 
E'H = r 
EH = s 



As we may expect r to reach its minimum near 6 = ± 90°, we first compute that 

value of the ratio — which satisfies our equation for this value of 6. From VR 2 — s 2 
e 

n _ 7D __ 

= 2e we obtain — = V&. For — < V5, as can easily be seen, the minima of r will 
e e p _ 

occur at values of \6\ < 90°, whereas for - > V5 the minima of r occur at |^| > 90°. 



6 " The position of the maxima and minima is evident. I therefore omit a discussion of the second differential quotient, which 

is a rather clumsy expression: 

d*r . , . „ ., «*{sin 0(cos * + cos 20) + sin 30 + sin 40} 2 
^f, = - £ (cos + 4 cos 20) - e 2 {cos 20 + 4(cos 30 + cos 40) - sin sin 20} Ri _ e , (ain ^ + slp ^ 



Willy Habtner 111 

The upper limit of <f> is defined by the equation sin cf> + 2 sin 2</> = 0, which is 

R — 

satisfied by \<f>\ = 104?477. Hence, for all — > V5, r will reach its minimum at 

90° < <f> < 104?477. e R 

In the following table, some corresponding values of (f> and — are given for which 
r becomes a minimum. e 



* 


R 




e 


90?0 


2-24 


101-0 


5-39 


102-0 


7-10 


103-0 


11-03 


104-0 


31-3 


104-477 


undefined 



It shows, in general, that — grows very rapidly near the critical value of c/> = 104-477, 

e R 

and, in particular, that for Ptolemy's value of — = 20, r reaches its minimum 

£ 

between <f> = 103° and 104°. Indeed, this statement has only theoretical interest 
because Ptolemy's and our own concern is not the minimum of the radius vector r, 
but that of the geocentric distance, which we shall henceforth designate by s (radius 
vector EH, in Figs. 9 and 10). Before entering upon a discussion of the latter quantity, 
however, I have to come back to my statement concerning the interchangeability of 

R 

the Ptolemaic curve with an ellipse, in the case of a sufficiently large value of — . 

e 

In Fig. 11, as in the preceding case, we have E = Earth, E' = punctum aequans, 
F = centre of a small circle with radius e = FE' = E'E, in whose circumference a 
point P revolves. The angle AFP which corresponds to a certain moment t will be 
called ip here. Availing ourselves of a well-known method, we construct the isosceles 
triangle FPS with apex P. By producing PS over S and making PK = R, the point 
K will evidently lie on an ellipse with the centre F and the axes R + e and R — e. 
For t = t , we choose tp = 0, so that K , on the apse-line, coincides with A. 

Our problem then is to express the radius vector E'K = p, in analogy with the 
preceding, as a function of the angle <f> = AE'K. In raising the perpendicular TK 

on A E, we recognize that cos \p — — ^-^ , and sin tp = — -. On the other 

hand, we have E + e R ~ e 

(R - e) 2 = KT 2 +TS*= P 2 sin 2 <f> + (R - ef cos 2 y>. 
By substitution we find 

(B - ef = p 2 sin 2 <f> + (|^) 2 (P cos cf> - ef 

or = P > (sin 2 + g^) 2 cos 2 ,) - 2„ (*=$ cos + * (§^)' 



112 



The Mercury Horoscope of Marcantonio Michiel of Venice 




II 
03 I *s 



&5 1 w 



bo 



Willy Hartner 



113 



The solution of this quadratic equation is given by the rather cumbersome 
expression 



± 



(R - e)V2 



2(R* + £ 2 — 2Re cos 2<f>) 



[V2(B 2 + e 2 )[(R + ef — e] + e\R — e) 2 — [42?e(22 2 + 2Re) - e{R — e) 2 ] cos 2cf>) 
e(R — e) 2 cos cf> 



+ 



22 2 + e 2 - 222 £ cos 2</> 



(3) 











-$ 


$ FOCUS OF 
* ELLIPSE 




















T 
S 






<p 














M 


$X 


\p 






\ • 




F "*/ 








\ / 
/ \ 
1 

> 


>fi 




V 
1 

; 


\ 




\ 






/ 


\ 




\ 






/ 


\ 




\ 






• 






>» 














V 





Fig. 11. Ellipse substituted for Ptolemaic curve 



where, of course, only the positive sign of the first term is applicable to our 
case. 



Unfortunately, there is no way to show that, for a sufficiently large value of 



R 



this formula for the radius vector p of the ellipse passes over into the one for r as 
given by equation (1). We therefore have to content ourselves with proving it 

R 

numerically for the Ptolemaic ratio — = 20. 



In letting e = 1, our formula (3) becomes 

19V2 a/353,241 — 34,839 cos fy 
P = ~2 401 — 40 cos 2cf> 



361 cos (/> 
40T — 40 cos 2(f>' 



(4) 



114 The Mercury Horoscope of Marcantonio Michiel of Venice 

With a very small error, this formula can be simplified as follows : 

19-10 2 a/2(35-324 — 3-484 cos 20) 9-00 cos 

9 ~ 2~~ ' 40(10 — cos 20) "" 10 — cos 20 

23-75A/70-65 — 6-97 cos 20 + 9-00 cos 

~ 10 — cos 20 ' 

In order to simplify the radicant 70-65 — 6-97 cos 2<f>, it seems practical to reduce 
it to the form c . (10 — cos 2<f>), where 7-065 > c > 6-97. Remembering then that, 
for = and cos = cos 20 = 1, the numerical value of p has to be 22-00, 
we obtain from 

23-75a/c(10 — 1) + 9-00 



22-00 



10 — 1 

63-00\ 2 

= 7-035, 



,23-75/ 

which, in fact, lies between the limits indicated. 
Hence we finally obtain the simplified formula 



63 . VlO — cos 20 + 9 cos 

D < — ' ; . .... (5) 

H 10 — cos 20 v ' 

It can easily be shown that the total error committed by this series of simplifications 
does not exceed 0-1 per cent. This means that, in computing the values of p to the 
second decimal place, the last digit may in extreme cases be affected with a maximum 
error of ± 2, for R = 20 and e = V°. 

Our Table 1 serves in the first place to compare r and p ; in the second, to show the 
variation of r in the neighbourhood of its minimum (between = 103° and 104°, see 
above). In the last column, the values of the geocentric distance s, in which we are 
particularly interested, are listed (three decimals in general, four for the interval 
116° < < 122°, to show the variation of s near its minimum, as well as the situa- 
tion of the latter). The table is computed for = 0, ± 5°, ± 10°, . . . , ± 180°, 
except for the critical interval 95° < < 125°, in which fall the minima of r and s. 
r and p are computed to the second decimal place. Only for 101° < < 106°, the 
values of r are given with a greater accuracy (three or four decimals). The geocentric 
distance is computed from s = V r 2 + e 2 + 2re cos 0. 

The table demonstrates with sufficient clarity that the "Ptolemaic curve" (about 

which no word can be found in Ptolemy's Almagest) is practically interchangeable 

with the ellipse as defined above, which it has been our aim to prove. The geocentric 

distance, s, occurs only as a denominator in the formula for the equation of the 

centre, — v 71 , 

. , , £ sin 
sin (0 — v) = -. (6) 

<s 

As this quantity, in Ptolemy's table of anomalies 72 , is given only to the minute of a 
degree, it makes practically no difference whether we compute s from the radius vector 
r, as in our table, or from the approximate value p, which corresponds to the ellipse. 

' Such an extreme case occurs for = 90°, where the difference "approximate minus exact" amounts to + 0-017; see note 
to our Table 1. 

71 4> = LAE'B. (Figs. 9 and 10) = mean anomaly; v = /_AEH = true anomaly.or apparent anomaly, according to Ptolemy 
(cf. footnote 56, page 106). 

" Almagest XI.ll (Vol. II, p. 265). 



Willy Habtneb 



115 



In order to demonstrate the low degree of accuracy attained by Ptolemy, and 
later by al-Battani 73 , and thus to exclude the last doubt as to the validity of my 

Table 1 



* 


r 


p 


r - p 


s 


<t> 


r 


p 


r —> 


s 


0° 

5 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

60 

65 


22-00 
21-98 
21-93 
21-82 
21-68 
21-51 
21-32 
21-10 
20-88 
20-63 
20-39 
20-15 
19-92 
19-71 
19-52 
19-34 
19-18 
19-06 
18-97 


22-00 
21-98 
21-93 
21-80 
21-65 
21-48 
21-26 
21-03 
20-82 
20-57 
20-33 
20-10 
19-88 
19-68 
19-48 
19-36 
19-18 
19-01 
19-00* 


0-00 
0-00 
0-00 
+ 0-02 
+ 0-03 
+ 0-03 
+ 0-06 
+ 0-07 
+ 0-06 
+ 0-06 
+ 0-06 
+ 0-05 
+ 0-04 
+ 0-03 
+ 0-04 

- 0-02 
0-00 

+ 0-05 

- 0-03 


23-000 
22-976 
22-916 
22-788 
22-623 
22-420 
22192 
21-927 
21-655 
21-349 
21-047 
20-740 
20-438 
20-153 
19-884 
19-623 
19-379 
19173 
18-996 


111° 
112 
113 
114 


18-90 
18-91 
18-91 
18-92 






18-562 
18-553 
18-545 
18-538 


115 


18-93 


18-97 


- 0-04 


18-535 


116 
117 
118 
119 


18-9455 
18-9578 
18-9711 

18-98527 






18-5290 
18-5253 
18-5227 
18-5211 


120 


19-00000 


19-05 


- 005 


18-5203 


70 
75 

80 
85 
90 


121 
122 
123 
124 


1901549 
1903171 
19-048 
19-064 






18-5176 
18-5212 
18-522 
18-524 


125 


19-084 


19-08 


0-00 


18-528 


95 


18-91 


18-98 


- 0-07 


18-850 




130 
135 
140 
145 
150 
155 
160 
165 
170 
175 
180 


19-18 
19-29 
19-41 
19-52 
19-63 
19-74 
19-83 
19-90 
19-95 
19-99 
20-00 


1917 
1929 
19-38 
19-52 
19-63 
19-73 
19-83 
19-91 
19-95 
19-99 
20-00 


+ 0.01 
0-00 

+ 0-03 
0-00 
0-00 

+ 0-01 
0-00 

- 0-01 
0-00 
0-00 
0-00 


18-553 
18-596 
18-655 
18-710 
18-771 
18-839 
18-893 
18-936 
18-966 
18-994 
19-000 


96 
97 
98 
99 


18-90 
18-89 
18-88 
18-88 






18-821 
18-794 
18-767 
18-750 


100 


18-88 


18-91 


- 003 


18-732 


101 
102 
103 
103-50 


18-8726 
18-8703 
18-8691 
18-8688 
18-8688 
18-8689 






18-707 
18-688 
18-669 
18-661 
18-656 
18-652 


103-75 
104 


* Exact value p 90 (e) = 18-978 
According to approximative formula p so lal = 18-995 


105 


18-8699 


18-91 


- 0-04 


18-636 


P.o ,a ' - P9o <e ' = + 0-017. 


106 
107 
108 
109 


18-8719 
18-88 
18-88 
18-89 






18-621 
18-607 
18-594 

18-582 




110 


18-89 


18-92 


- 0-03 


18-572 













statement, I give in Table 2 a comparison of the theoretical values of the equation 
of the centre, <f> — v, computed from the above formula (6), for e = 1, with those 
found in Ptolemy's and al-Battani's tables (columns 1-4). For all A =t= 0, the 
values of s corresponding to Ptolemy's wrong figures are computed from the latter 
and compared with those of Table 1 (columns 5-7). 

As is seen, the minimum change in s caused by an error of 1' in (<f> — v) amounts to 
O08e, whereas we have found that, on the other hand, the maximum error committed 
by substituting the ellipse for the Ptolemaic curve was 0-07e (attained only twice: 
for <f> = 35° and <f> = 95°, cf. column "r — />" in the preceding table). 

73 Al-BattAnI's tables (Op.Astr., II, pp. 132-137) still have the same degree of accuracy as Ptolemy's, but are more elaborate ; 
they list the equation of the centre for every degree, while Ptolemy, for 0° < <t> < 90°, lists it only from 6 to 6°, and for 
90° < <t> < 180° from 3 to 3°. The sum of Ptolemy's third and fourth columns is equal to al-Battani's third column. A 
difference of ± 1' occurs only in a few cases, and only one such case appears in our list, viz. <t> = 165°. In order to avoid 
interpolations the figures in column 3 of my table are taken from al-BattanI. 



116 



The Mercury Horoscope of Marcantonio Michiel of Venice 



We now have to remember (cf. above, Section 9 (b)) that Ptolemy's really ingenious 
theory of Mercury owed its origin to the circumstance that, in comparing the maxi- 
mum elongation of Mercury (from the Sun's mean longitude), he had found that this 



Table 2 







<t> - 


- V 




A 


s 




(5 


<t> 










(Ancient 






(Ancient 










minus 






minus 




Theoretical 


Ancient* 


Theoretical) 


Theoretical 


Ancient 


Theoretical) 


0° 


0?000 = 0° 


0' 


0° 


0' 


0' 






0-00 


5 


0-217 = 


13 





15 


+ 2 


22-98 


19-98 


- 3-00 


10 


0-435 = 


26 





28 


+ 2 


22-92 


21-31 


- 1-61 


15 


0-651 = 


39 





40 


+ 1 


22-79 


22-23 


- 0-56 


20 


0-866 = 


52 





53 


+ 1 


22-62 


22-19 


- 0-43 


25 


1-080 = 1 


5 


1 


5 





— 


— 


0-00 


30 


1-291 = 1 


17 


1 


17 





— 


— 


0-00 


35 


1-499 = 1 


30 


1 


31 


+ 1 


21-93 


21-66 


- 0-27 


40 


1-701 = 1 


42 


1 


43 


+ 1 


21-66 


21-45 


- 0-21 


45 


1-898 = 1 


54 


1 


54 





— 


— . 


0-00 


50 


2-086 = 2 


5 


2 


6 


+ 1 


21-05 


20-91 


- 0-14 


55 


2-264 = 2 


16 


2 


16 





— 


— 


0-00 


. 60 


2-429 = 2 


26 


2 


25 


- 1 


20-44 


20-54 


+ 0-10 


65 


2-572 = 2 


35 


2 


35 





— 


— 


0-00 


70 


2-709 = 2 


43 


2 


43 





— 


— 


0-00 


75 


2-821 = 2 


49 


2 


49 





— 


— 


0-00 


80 


2-913 = 2 


55 


2 


54 


- 1 


19-38 


19-47 


+ 0-09 


85 


2-978 = 2 


59 


2 


58 


- 1 


19-17 


19-25 


+ 0-08 


90 


3-018 = 3 


1 


3 


1 





— 


— . 


0-00 


95 


3-029 = 3 


2 


3 


2 





— 


— 


0-00 


100 


3014 = 3 


1 


3 


1 





— . 


— 


0-00 


105 


2-971 = 2 


58 


2 


58 





— 


— 


0-00 


110 


2-900 = 2 


54 


2 


54 





— 


— 


0-00 


115 


2-803 = 2 


48 


2 


49 


+ 1 


18-54 


18-44 


- 0-10 


120 


2-680 = 2 


41 


2 


41 





— 


— 


0-00 


125 


2-534 = 2 


32 


2 


32 





— 


— 


0-00 


130 


2-366 = 2 


22 


2 


22 





— 


— 


0-00 


135 


2-179 = 2 


11 


2 


11 





— 


— 


0-00 


140 


1-975 = 1 


59 


2 





+ 1 


18-66 


18-42 


- 0-24 


145 


1-757 = 1 


45 


1 


47 


+ 2 


18-71 


18-43 


- 0-28 


150 


1-526 = 1 


32 


1 


32 





— 


— 


0-00 


155 


1-285 = 1 


17 


1 


18 


+ 1 


18-84 


18-63 


- 0-21 


160 


1-037 = 1 


2 


1 


3 


+ 1 


18-89 


18-66 


- 0-23 


165 


0-783 = 


47 





48* 


+ 1 


18-94 


18-54 


- 0-40 


170 


0-525 = 


32 





32 





— . 


— . 


0-00 


175 


0-263 = 


16 





16 





— 


— . 


0-00 


180 


0-000 = 














— 


— 


0-00 



* "Ancient" = Ptolemy and al-Battani, whose figures are identical for the values of <t> listed, 
except for 4> = 165°, where Ptolemy has <t> — v = 0° 47', whence A = 0. 

planet, in the course of one tropical revolution, passes twice through a perigee 
position, whereas it has only one apogee. 

He refers to the following two pairs of corresponding observations 74 : 

2nd February, 132 a.d. (mean long, of Sun = Aquarius 10° = 310° 

= mean long, of centre of epicycle) : 

max. east elong. = 21° 15' 

2nd February, 141 a.d. (mean long, of Sun = Aquarius 10° = 310°): 

max. west elong. = 26° 30' 



47° 45' 



Almagest, IX.8 (Vol. II, p. 139f). 



Willy Hahtner 



117 



II ^ 



4th June, 134 a.d. (mean long, of Sun = Gemini 10° = 70°): 

max. west elong. = 21° 15' 

4th June, 138 a.d. (mean long, of Sun = Gemini 10° = 70°): 

max. east elong. = 26° 30' 



47° 45' 



This means that, in both perigee positions, the angle subtended by the epicycle is 

47° 45', and hence the angle subtended by its radius, a P = 23° 52' 30", whereas in 

the apogee position the maximum elongation is found to be a A = 19° 3'. 

With the aid of elementary trigonometrical considerations Ptolemy then contrives 

to prove that his theory tallies perfectly with his observations if one sets e = 3 P , 

and the radius of the epicycle, q = 22 p 30' (for R = 60 p ), or R = 20 and q — 7-5 

(for e = 1). Accepting Ptolemy's assumption that the longitude of the apogee, 

about 140 a.d., is ca. Libra 10° = 190°, the anomalies of the centre of the epicycle in 

the above cases will be 120° and 240° respectively. In both cases we have s = 18-5203 

(see Table 1), whence 

7.5 
sin g p 



18-5203 



0-404961; <y, 



23?8887 = 23° 53' 19" 



For the apogee we find 



7-5 



sin a A = 



23-0000 



(Ptolemy 23° 52' 30", error < 50"). 

0-326087; a A = 19?0314 = 19° 1' 53" 

(Ptolemy 19° 3', error < 70"). 



The agreement of theory and observation thus must be said to be extremely good. 
As regards the value of </> for which the geocentric distance becomes a minimum, it 
appears that Ptolemy naively assumes it to be i 120° exactly. The same naive 
assumption is still found thirteen centuries later in Peurbach's Theoricae Novae 
Planetarum. This however is not true, although it is a very close approximation. I 
refer to our Table 1, where the lowest value of s listed (18-5176) corresponds to 
<f> = 121° 75 . 





v 








\°2 






/ 


\ 
\ 








\ 


6 




J*r-~~ 


/w 






/w; 


^\ 




/ 9 / 




Je^ 


k\ ^V \ 


I /<*\ 




v vl 





Fig. 12. Determination of minimum geocentric distance (ellipse) 



75 It is, of course, of no practical interest to compute the values of s and <f> for the theoretical minimum. In any case, this 
would involve rather elaborate considerations and calculations. That the ellipse substituted for the Ptolemaic curve, which 
indeed offers itself to direct mathematical treatment, does not furnish a satisfactory answer in this case, may be seen from 
the following, illustrated by Fig. 12. 



118 The Mercury Horoscope of Marcantonio Michiel of Venice 

As mentioned before, the first European author I know of who expressly stated the 
similarity of the curve described by the centre of the epicycle with an ellipse was 
Peurbach, and even he contents himself with saying that it is a "kind of oval". 
In the Islamic world, however, as will be seen, the discovery is of a much earlier 
date; to prove this, I refer to the Libros del Saber de Astronomia 76 , composed in 
1276-77 by order of King Alphonse X of Castile, where we find all necessary informa- 
tion in the section bearing the title Libros de las laminas de los VII planetas 17 . This 
section consists of Spanish translations of two western Arabic treatises, one (Book I) 
by Abu 'l-Qasim Asbagh Ibn Muhammad Ibn al-Samh of Cordova (d. 1035), the 
other (Book II), which was written ca. 1081, by Ibrahim Ibn Yahya al-Naqqash 
Abu Ishaq Ibn al-Zarqali, commonly cited by his latinized name Azarquiel 
(d. 1100). The problem dealt with is the same in both treatises, namely, to construct 
instruments {laminas = "disks") reminiscent in their general aspect of planispheric 
astrolabes, and consisting of a series of graduated circles or rings, which permit the 
quick determination of the position of the planets without computation 78 . 

Ibn al-Samh, who constructs one particular disk for each of the seven planets, 
devises two different constructions for the motion of Mercury, one very crude based 



(Continuation of footnote 75 from page 117) 

Fig. 12 shows an ellipse AiBA^, with centre F, foci ®! and 2 , the major axis 2a, and angle of eccentricity O^-F = w. As in 
Fig. 11, E (Earth) marks a point between O, and F, whose distance from F is 2s. Our problem is to And the minimum distance 
EK = s and the corresponding angle A 2 EK = v (apparent anomaly), from which, finally the angle A a E'K = <t> (mean anomaly) 
has to be computed. 

s being the normal through E by which Z.O x -K0 2 = y is bisected, we have 

*,£ _ sin (180°^ v) _ ship = O.g 
y ~ <b 2 E 
2 









<biE . y 
sin- 


or, calling <£>i-K = 


= q, <J> 2 2 = 


= 2a 


- q, *i® 2 = 2/, <b y E = p 

p - 2/ q - 2a 
p q 


whence 






2/ 2a q 
— = — or - 
p q a 



V 

'/• 

In other words, the perpendicular raised in E on A t A 2 will meet O^ = a in a point Q t such that O^ = q; it will similarly 
meet the line 2 iJ produced in Q 2 , where 9 2 Q 2 = 2a — q. 

By introducing l_w = ^,.BF = O^i-B, we can write the last relation 

2e 

q = a : 

sin iv 

2e 
and, correspondingly, 2a 

From q 2 + (2a - qf - 2q . (2a - q) . cos y = (2/) 2 

we then obtain by substitution 

a 2 sin 2 w(l — 2 sin 2 w) + 4e 

cos y = „ . , 

a 2 sin 2 w — 4e 

19 
In the present case (cos w = ~- , w = 25?2087, a = 21, e = 1) the computation yields 

cos y = 0-7233593 and y = 43°6675. 

. V 
sin- 

Then from sin v = ^^ 

we obtain (v) = 60?8340 and v = 180° - (v) = 119»1660. 

Finally, the formulas = P ' Sm - yields (a being = 97S3322): * = 18-5189. 
. y 
sin - 

By transforming the formula for the equation of the centre, sin (<t> — v) = , 

we obtain ctg 4> — ctg v -. , 

e 8 sin D 

which furnishes 4> = 1219797 

as the anomaly for which the "elliptic" geocentric distance, s w , becomes a minimum. 

According to the formula of the Ptolemaic curve (1), however, we obtain for this value of 4>, r = 190283597 and for the 
"Ptolemaic" geocentric distance s„ = 18-5210, which is larger than the smallest value listed in our table (approximate minimum, 
s m = 18-5176). 

Thus the minimum of s E is a little larger than that of s p and occurs at a value of \<t>\, which is about 1° larger than that 
corresponding to the minimum of s„. 

76 Libros del Saber de Astronomiadel Rey D. Alfonso X de Castilla, ed. M. Itico Y Sinobas, 5 vols, in folio, Madrid, 1863-67. 

" Vol. II, pp. 241-284. ,8 Cf. G. Sarton, Introduction to the History of Science, Vol. II, Baltimore, 1931, p. 837, No. 14. 



Wllly Hahtskr. 



Ill) 



on a fixed circular eccentric (Ch. VII, p. 263), and one elaborate (Ch. XV, p. 2(VA), 
saying twice that it is grieite de fazer, i.e. "hard to make"). This latter, as far as I 
can see from the obviously mutilated text, is mainly based on Ptolemy's theory as 
described above and shows no feature of particular interest. 



■wjt « 




/- Av /"^.^tv-y- ■■■■'■■■■• 







: i lc< 



w : 



Fig. L8, Libru* tii't Saber, vol- III, p. 281 : fcha circles of Venus and of the superior planets 

Azarquiel's treatise, however, as far as Mercury is concerned, is of first-rate 
importance. Unlike that of his predecessor, his device serves the purpose of repre- 
senting the motions of all the seven planets on the two surfaces of one and the 
same disk, namely Venus and the superior planets on one (see Iftg. 13), and the 
Hun, the Moon, and Mercury on the other (see Fig. 14). Unfortunately, the drawings 
are as incorrect as they are beautiful, and as the text- accompanying them also bristles 
with errors it is often no easy task to make out the true meaning of the author. 
Yet from the wording of the chapter referring to the "Circles of Mercury" (Book II, 
Ch. IX, pp. 278-80), it is perfectly clear in what way Azarquiel wants those "circles'' 
to be drawn. 

It is, of course, the elliptiform curve with the axes 90 and 7<i mm m the middle of 
Fig- 1 4 that evokes our curiosity. It has been paid attention to by earlier historians 79 , 
but those I know of apparently failed to recognize its historical context and true 

19 R. \Voi>f, Oachichlt tier Astroiuimie, Munich. 1877, who mention* it on |>. 2U7, mid* thir remark: ". . . uutl w'eiH) 08 audi 
etvrua gewagt erachetnt, in dieser Ellipse, derail MUtelpitntt das Zeicheu der Sonne zvigt, clmm Vorlftufer tier Kcpler'adien 
Kllt]i.*i-n ftelit-n zu wollea, so doeimientirt *ie dagegEn, vie MSdler rlvhtig itervoriiebt, 'dasa man ec-lion frdh dfe CnuiogUchkeit 



1-20 



Trio Mercury Horoscope of Man-antonio Michiel of Venice 



significance* 1 *. The text of Chapter IX, however, which will be summarized here, 
leaves no doubt that it is nothing but the curve resulting from Ptolemy's theory, 
which wc discussed above in detail: 



I 
















P 



& 

1 1 i;; 

f 3 =: 



* 



3 



H.,. 



*. *\v 



^ 



>- 



Pkr. II. 



LibroH del Saber, vol. Ill, p. 2H2: the circle:* of Sim ami Moon, 
and tlir ell 1 1 iti form curve of Mercury 



Mark a point (our point fr\ see Fig. {)} which is 4P 42' distant, from the centre 
of the disk (/£) and make it the centre of a circle with the radius 2p 21' (s), and 
call this latter the "deferent of the centre of the deferent of Mercury" (el qerco 
huador* 1 del cerdro dd leitador de. msrenrio), Then draw a ''hidden" circle (*,e. 



Continuation of footnote fS J'n>i>< pag> ' t$) 

etngesehen bat, mil. dam exoentrl&ohen Kfelno In alien Ifillen auiuureichen'." As demonstrated above, Mildler'a "aelmn frllli" 
haft to ho interpreted us. "from the very bcnintLlin?", because (lie iJllplifnrm carve exists, at least, by implication, already in 

Ptolemy, 

Saktcin, lutrmturtiun 1 1, j>, sis: "A lljjni'e of the deferent of Mercury in the form of an ellipse with what look* like :i Sun in 
the centra la tin* second treatise an the plates of the fteveii planets {no. 14 in my list above, Madrid edition, vol. 3, 282. I £(145 in 
purely occidental: it in not In any sense) an anticipation of KKPLEk'£ diBoovery of the elliptieity of celestial orbitft", Josft M\ 
Mim.as V&Uiioaosa, BttwHot to&n hittoria <(/• (a etaieta ttpa&ola. Barcelona, 1 MO, p. 137. footnote 4ft, n't/ens to hi* treatment 
of the Lit/fua id loa himimty de toi ttfate tiUuii-itm, In Chapter X of ua BttaiMoa tobre Axnveu&&, then in print. Aft I clu not jkjhihch* 
this work, 1 can [!ii!> refer bo i«. SasoosIs review, In f*£f, 42. p. tfiSf, where a diseussiun of ttoelhpH is not mentioned, N" 
negativf eonolnalon. however, run he drawn from Ihis urgumeiktxm etSm^h. 

* D Note adtit'd in 1'ftnif. It wu oid.^ 1 alter having concluded the present study that The author found h Dorreol tnterpretatton 
of the curve ici qufstlon In Ai.khkd Whornkr. "I>1< j antroiionilHchen Werke Aifons X", in BUMolbtta MaHtfmafka, S. Polge, 
ti. Baud, Leipzig, lHUo, Mi 1 - I^S' Hf>. Tht'it', on pp, 150 61, Weoen'ER demonstrate-s its (.-onstrurtlon Hcruniinu In Ptoleniy't 
tiii-orv (his Fi^. 2, p. 157), though wltlmul i-nti-rlng upon n Tualhcruaiical analysis and fomnarfeon with tlie ellipfttr. lln then 
refers to BRasmTB fUDKBOLDlB roinm^ntjiry on PBVKBAta (1642), I'HSIISICS (WrRSTEFSEN', 15(38), and Rruinia'fi Aluusifcsirtw 

Novum (1351 ). BAB90B tltUrodaeUon II, p. n-tl ) emphasized the Importance ofWRGRsnsn's study, yet did not mention its most 
important |iolnt (cf, the ]ire(«dfiiK footnote). Weoeskr already points out the really amazing errors committed, and the 
t'anla*tir IheOtfH pUl forward ill tllU I'OIincrtlon b>" eattler IliBtorianS, particularly N. BSXZ, Gexchirhte drr HuhiibfsUmwtnu DM 
Flaitetott and Ktimelen, Leipzig, lsiS7 94. " Old Spanish lerador = "deferent", cf. modern Spanish tftrnr "to carry". 



Willy Hartner 121 

which will later be erased) with radius 8 IP round F and divide it clockwise into 
seventy-two equal parts, starting from the apse-line. Moreover, draw another 
"hidden" circle with radius 80P round a point on the apse-line (E') which is 
2p 21' distant from E, divide it counter-clockwise into seventy- two parts, and call 
it the "circle of equal motion of Mercury" (el cerco dell yguador del mouimiento de 
mercurio). Then transfer the division of the large circle round F on the small 
concentric circle and, on the other hand, draw the radii (as "hidden lines") from 
E' to the seventy-two divisions (cinculares, i.e. sections comprising 5°) of the 
circle round E' . Thereafter, mark the point on the apse-line which is 49P 21' 
distant from the point of intersection (M) of the small circle round F with the 
apse-line, and, placing subsequently the point of your compasses with a constant 
opening of 49P 21', in the subdivisions of the small circle, mark the points of 
intersection with the corresponding radii through E' . Finally, join every three 
of the points thus marked, by an arc, and there will result a figura pinnonada 
(a curve similar to a pignon). Et quando fizieres los cercos de mercurio assi 
cuemo te mostre aqui en este capitolo. salirtd so logar con ellos Men cierto. mas 
que dotra manera. ("And when you have made the circles of Mercury as I have 
shown here in this chapter, its position will result from them very accurately, 
more so than in any other way".) 

Thus the first explicit description of the curve of Mercury's true deferent, as well 
as its practical application, is undoubtedly Arabic; it precedes the first European 
mentioning by nearly 400 years. As Ibn al-Samh, half a century before Azarquiel, 
does not yet mention it, we may moreover safely assume that Azarquiel himself is 
entitled to the honour of having discovered it. 

The only difference between Azarquiel and Ptolemy lies in the change of the 

R 
numerical values of the parameters. Instead of Ptolemy's e = 3p and — — 20, 

R £ 

Azarquiel lets e = 2p 21' and — = 21. The question whether this was due to new 

e 

observations cannot be decided upon off hand : (al-Battani still has the Ptolemaic 
values). The radius of the epicycle, however, is not changed: Azarquiel makes it 
18P 30' for his major semi-axis of (49P 21' + 2p 21') = 5 IP 42', which corresponds 
almost exactly to Ptolemy's 22p 30' for a major semi-axis of (60 + 3) = 63P. 

Concerning the plate illustrating Azarquiel' s text, it may be well to note that it 
was obviously not carried out in accordance with the author's prescription. The 
small circle in the middle (which looks like a Sun and, therefore, has deceived many 
interpreters) is nothing but the small circle with radius e round F. The rays appar- 
ently emanating from it indicate its division into equal parts, but there are not 
seventy-two rays as prescribed, but only sixty. In the original, e is 2-5 mm, whence 
the major semi-axis should be 22 x 2-5 = 55 mm, and the minor, approximately 
20 X 2-5 = 50 mm, whereas we have stated above that they actually are 45 and 
38 mm, respectively (the outer, not the inner, ring counts, hence the different values 

given by Wolf and Madler: 82 and 67 mm for the whole axes). From R + e = 45 

-p 

and R — e = 38 results R = 41-5, e = 3-5, and — = 11-85, which is completely 

e 

wrong and must yield false results. Thus, evidently, the curve was not constructed 

correctly with the aid of the small circle, but probably drawn at random, at the 



122 The Mercury Horoscope of Marcantonio Michiel of Venice 

R 
whim of the designer. For — ^ 12, moreover, the curve would no longer have 

£ 

(practically) two axes of symmetry, as in the figure, but be noticeably narrower in 

its lower part; cf. the dotted curve corresponding to — = 10 in Fig. 10. 

e 

Finally, the orientation of the apse-line of Mercury is as wrong as it can be, viz. 
25° Pisces instead of ca. 24° Libra (Ibn al-Samh's text, Book I, Ch. XIII, p. 262, 
has 23° 40' Libra for the epoch 416 a.h. = 1025 a.d.). 

Considering the general dependency of Peurbach on Arabic astronomy (cf. 
below, Section 12), it is highly improbable that his statement concerning the oval 
shape of Mercury's deferent should have no connection with Azarquiel and the 
Libros del Saber. But it will not be easy to find out through what channels he got 
acquainted with the achievements of his Arabic predecessor 82 . 

There is, of course, no doubt that Copernicus as well as Kepler were thoroughly 
conversant with the contents of Peurbach's treatise, and perfectly aware of the 
oval shape of Mercury's deferent. But in spite of this, Copernicus did not think of 
introducing ovals or ellipses into his heliocentric system 83 . And, certainly there 
appears to be no natural or obvious way leading from the oval (or elliptiform) 
geocentric deferent to an elliptic heliocentric orbit. As for Kepler, it is not impos- 
sible that the first idea of using ellipses may have come to his mind through Peur- 
bach's treatise. As a matter of fact, his first attempt, after having proved the 
impossibility of a circular orbit, was to introduce an oval curve which was wider near 
the apogee, and narrower near the perigee. Yet the independence of his procedure 
is so striking that even the definite proof of one or the other impulse received from 
others would not impair the greatness of his discovery 84 . 

Another question, however, is the later development of instruments similar to 
those of Azarquiel. It would be worth while investigating whether the planispheres 
of Peter Apianus 85 , which he constructed for the same purpose (Kepler called his 
efforts an "industria miserabilis"), show a definite dependence on the ones described 
in the Libros del Saber. In particular, it would be interesting to know if he, too, makes 
use of an ellipse in connection with Mercury. An answer to both questions can easily 
be found in his Astronomicum Caesareum of 1540. I should expect it to be to the 
affirmative. 

11. Mathematical Theory Versus Physical Reality. The Nature 
op the Spheres According to Alhazen 

In the early fourth century B.C., Eudoxus of Cnidos devised his theory of homo- 
centric spheres which, to the best of our knowledge, is the first attempt made in 



82 Here reference is also made to the experiences made in an early Islamic zodiacal interpretation: Arthur Beer, "The 
Astronomical Significance of the Zodiac of Qusayr'Amra", in K. A. C. Creswell's Early Muslim Architecture, Vol. I, pp. 296- 
303, Clarendon Press, Oxford, 1932. (See also F. Saxl, I.e., pp. 289-295.) 

83 As for Copernicus, the famous passage about the ellipse cancelled by the author's own hand (Be Revolutionibus, Book III, 
Ch. 4, original Ms., fol. 75r, German translation by C L. Menzzer, Thorn, 1879, p. 139) is not, as claimed time and again (for 
instance, by J. Hopmann, in his Preface to the anastatic reprint' of Menzzer's translation, Leipzig, 1939, p. VII) an anticipation 
of Kepler's discovery, but only a mathematical triviality. Those wishing to maintain the myth are recommended to read the 
whole passage, and not only the five or ten lines in question. It would help them to understand that Copernicus does not 
refer to planetary motion at all. In V.4 (ed. Menzzer, p. 272) a deviation from the perfect circle is mentioned in connec- 
tion with the planets, but it is not said that the resultant curve is an ellipse. Kepler discusses this chapter extensively in De 
motibus stellae Martis, 1.4, q.v. 

84 E. ZlNNER, Oesehichte und Bibliographie der astronomischen Literatur in Deutschland zur Zeit der Renaissance, Leipzig, 
1941, p. 33, states that Erasmus Reinhold (1511-43), "the best theoretician of his time, by referring to the oval orbit of the 
centres of the Moon's and Mercury's epicycles, initiated or adumbrated ('hat angebahnt') future considerations". Zinner does 
not mention the passage found in Peurbach's treatise. It is, of course, the source of Reinhold'S statement, as he wrote himself 
a commentary on it (published in 1542). 

85 See R. Wolf, op. cit., p. 265. 



WnviW TTahtnkh 



123 



Greek antiquity at subjecting the complicated motions of the planets to mathematical 
treatment. From the reports on which we have to rely it. seems to result that 
Eudoxuk himself did not bother about the physical reality of his construction (to 
caeh planet he attributes a series of spheres, one revolving inside the other around 



t 













tl 



Fig, 15, A page from Al-[1Iiwx!'s Mr.iJi'adie G(fflQW Ms. Berlin Qu. Mil'S. Po], L90r. 

Arabic text with two inserted sketches; the upper showing tho iiintimi of Voiiiih and the 

superior planets, awl the lower that nf MWrury 



different axes, the planet itself being fixed to the equator of the innermost sphere, 
while the outer sphere, in every case, was thought to have the same rotation as that 
of the fixed stars). 

It was Aklstott.e who tried to raise this theory of homoeentric spheres to the rank 
of a system based on physical reality, by inserting sets of reacting spheres between 
the single, and originally independent, planetary systems so as to reduce the complex 



124 The Mercury Horoscope of Marcantonio Michiel of Venice 

motion of the innermost sphere of one planet to the simple motion of the outermost 
sphere of the next one. 

In the Almagest the question of physical reality is not discussed. Although in 
Greek the word o<j>aipa may refer to the solid sphere as well as the circle, Ptolemy 
in his Almagest uses it only to denote the total of superimposed circles (kvkXos, 
€ttikvk\os) representing a planet's orbit, and thus the orbit itself 86 . In his Hypotheseis, 
however, he takes a different attitude; there, obviously, the term sphere has to 
be understood in our modern sense of the word, i.e. as referring to solid spheres or 
spherical shells. 

During the Islamic Middle Ages we encounter both interpretations, varying 
according to the greater interest of the authors in mathematical and astronomical, 
or in physical and philosophical matters. Thus while astronomers such as 
al-Battani and al-Biruni 87 show but little interest in the physical reality of their 
system, the question plays a predominant part in the writings of physicists, among 
whom I mention first and foremost the great Alhazen (al-Hasan Ibn al-Husain 
Ibn al-Haitham of Basra, c. 965-1039); in the late Middle Ages, probably because 
of the enormous reputation of Alhazen's writings, the physical interpretation obtained 
a footing also in certain treatises of a purely astronomical character. We find it not 
only in Qazwini's famous Cosmography 88 but also in an astronomical treatise by 
al-Jaghmini 89 . The latter seems to have been highly esteemed in Islamic countries. 
There exist numerous manuscripts of his treatise ; it was translated into Persian and 
commented upon by the Persian astronomer and philosopher 'Ali Ibn Muhammad 
al-Jurja.ni (1340-1413). The dependency of early Renaissance astronomers on 
Alhazen and al-Jaghmini is beyond doubt. Yet I am unable to tell at the moment 
from which of the two (possibly from both), and through which channels, they drew 
their information. _ 

The theory set forth by Alhazen in his treatise Fl hai'at al-'alam ("On the Shape 
of the Universe") 90 is in its main outlines as follows: 

The universe, which is spherical in shape, consists of nine spherical shells 
(afldk, plural of falak = o-^alpa) 91 gliding inside one another; each of these 
shells again is composed of a set of concentric or eccentric shells or complete 
spheres. There is no empty space or "void". The Earth with its water is sur- 
rounded by air, which again is surrounded by fire. The sphere of fire is limited 
by that of the Moon, after which come the spheres of the six other planets and of 
the fixed stars. Finally, the outer limit of the universe is the "sphere of spheres" 
(Caelum Empyreum). While the four elements of which consist the earthy, 
watery, airy, and fiery spheres are either heavy or light, the fifth element 
(Arabic aithir = Aristotle's "aether"), which is the matter filling the trans- 
lunar world, is neither the one nor the other. Unlike the four terrestrial elements 
it has the essential quality of eternal circular motion. 



86 Thus, in III. 3 (ed. HEIBERG, I, p. 216, 1-12) he speaks about the Oetreis k*1 ra^eis ruv Ivtous <r<pa.ipu'.s abrwu kvkXoiv 
("the situations and positions of the circles lying in their spheres"). 

87 For al-Biruni's (973-1048) purely mathematical treatment, see Fig. 15, showing a page from one of the earliest manu- 
scripts known of his Mas'Mic Canon (al-QInun al-Mas'udi), written less than a century after the author's death. 

88 Abu Yahya ZakarIya Ibn Muhammad Ibn Mahmud al-QazwinI, born 1203/04, d. 1283. Arabic editions of his Cosmography 
by F. WUstenfeld (2 vols.), Gottingen, 1848-49: German translation of the first part of Vol. I (containing the astronomical 
part), by H. ETHE, Die Wunder der Schopfung, Leipzig, 1868. 

, 8 » Mahmud Ibn Muhammad Ibn 'Umar al-Jaghmini (d. 1344/45), see G. Kudloff and A. Hochheim, "Die Astronomie des 
Gagmini", in Zeitschrift der Deutsehen Morgenlandischen QeseUschaft, Vol. 47 (Leipzig, 1893), pp. 213-275. 

,0 See K Kohl, "tiber den Auf bau der Welt nach Ibn al Haitam", in Sitzungsberichte d. Physik. Med. Sozietat in Erlangen, 
Vol. 54/55 (1922/23), Erlangen, 1925 pp. 140-179. 

91 The ambiguity of the term falak, which is exactly the same as that of o-cpaipz, is expressly discussed in the beginning of the 
3rd part. In Alhazen's practical application, however, it nearly always denotes the globe or the spherical shell, not the circle. 



Willy Hartner 



125 



(a) In the first place, following Alhazen's example, I shall describe the system of 
the Sun, for the sole reason that it is simpler than the others (see Fig. 16 92 ). The 
sphere of the Sun is a corporeal (material and perfectly transparent) spherical shell 
concentric with the centre of the universe (the Earth). It surrounds the sphere of 
Venus and is itself surrounded by that of Mars. In this spherical shell, called al-falak 
al-mumaththal = "assimilated (i.e. concentric or parecliptic) sphere", another such 




Fig. 16. The spheres of the Sun 



E = Earth 

A = apogee of Sun's centre 
B = perigee of Sun's centre 
F = centre of eccentric sphere 



a = concentric sphere 
b = eccentric sphere 
S = Sun 



shell is embedded eccentrically, in such a way that its interior surface touches from 
outside the interior surface of the first (concentric) shell, while its exterior surface 
touches from inside the exterior of the concentric one. This second shell, called 
al-falak al-kharij al-markaz = "eccentric sphere" (which in this case is identical with 
al-falak al-hamil = "bearing sphere" or "deferent") rotates from west to east round 
an axis through the poles of the ecliptic, carrying along the Sun, which is a solid 
globe fitting exactly between the two surfaces of the eccentric shell. The apogee 
(Arabic awj, whence mediaeval Latin aux, gen. augis, as consistently used by 
Pburbach 93 ) precedes the point of summer solstice by 24|° (i.e. has a longitude of 
65i°) 94 . 

(6) Before entering upon the discussion of the problem of the planet which has our 
particular interest, it will be useful to review the less complicated system devised by 
Alhazen to describe the motion of Venus and the superior planets. 

The sphere of Venus is embedded between those of Mercury and the Sun. It con- 
sists of a spherical shell concentric with the Earth and an eccentric shell analogous 



82 After Kohl, I.e., Fig. 2 (p. 154). 

93 The Libros del Saber, too, employ the terms auxe or alaux. 

94 Alhazen here gives the same value as Ptolemy, Almagest, III. 4 (Vol. I, p. 167), although al-Battani some time before had 
discovered the displacement of the apogee. Yet in another place he refers to the new discovery. 



126 



The Mercury Horoscope of Marcantonio Michiel of Venice 



to that of the Sun (see Fig. 17). Between the two surfaces of the latter a rotating 
(solid) sphere (the sphere of the epicycle, falah al-tadwlr) is embedded, in whose 
equator the spherical body of Venus is fixed. 

The concentric sphere has a slow rotation (in the direct sense) of 1° in 100 years 
(Ptolemy's value of the precession of the equinoxes; i.e. the apse-line, in the case 
of all planets, is considered immovable relative to the fixed stars 95 ) ; the rotation 
takes place in the plane of the ecliptic. 

The eccentric, or bearing, sphere rotates around another (inclined) axis, likewise in 
the direct sense, making one whole revolution in the course of one tropical year 96 . 




Fig. 1 7. The spheres of Venus and superior planets 



E = Earth 

A = apogee of centre of epicycle 
B = perigee of centre of epicycle 
F = centre of eccentric sphere 
E' = punctum aequans 



a = concentric sphere 
b = eccentric sphere 
e = sphere of epicycle 
H = centre of epicycle 
T = planet 



The rotation is uniform when seen from the punctum aequans, E' , which lies on the 
apse-line beyond the centre of the eccentric, its distance being twice that of the 
latter from the Earth. 

The sphere of the epicycle, again, turns in the same sense around an axis that is 
inclined to that of the concentric as well as to that of the eccentric. Its period is that 
of the planet's synodic revolution. 

(c) After this, there will be no difficulty in understanding the system of Mercury. 

Again, the sphere of Mercury (Figs. 18 and 19) is embedded between those of the 
Moon and Venus. In perfect analogy with the preceding, the concentric sphere moves 
(directly) 1° in 100 years, in the plane of the ecliptic. A first eccentric, called al-falak 
al-mudir, "the turning sphere", is embedded in the concentric, while a second 
eccentric, the deferent proper (al-falak al-hamil), is embedded in the first. For the 
centres of these three spherical shells I refer to our Figs. 9 and 10. In the initial 
position (centre of the epicycle in the apogee, see Fig. 18) the three centres, E, F 



95 Cf. above, Section 9 (a). . it A . ^ ... x ... * ... J c .. . i * «, 

'• True also in the case of Mercury. For the other planets, the time of one complete rotation of the deferent is equal to the 
planet's tropical revolution. 



Willy Hartneb 



127 



and M lie on the line which joins the apogee with the Earth, called by us "apse-line" 97 . 
For other positions of the centre of the epicycle (see Fig. 19), the centre of the 
deferent (P in Figs. 9 and 10) lies on the circle with radius e round F. Again E', 
situated, in the case of Mercury, in the middle between E and F, marks the punctum 
aequans from which the revolution of the centre of the epicycle appears uniform. 

The sphere of the epicycle carrying the planet itself in its equator is fitted between 
the two surfaces of the deferent. 




Fig. 18. The spheres of Mercury, initial position (<f> = 0) 



E = Earth 

E' = punctum aequans 
F = centre of turning sphere (Mudir, 6) 
M = centre of deferent (Hamil, c) 
A = apogee of centre of epicycle 



a = concentric sphere 

b = first eccentric sphere or turning sphere (Mudir) 
c = second eccentric sphere or deferent (Hamil) 
e = sphere of epicycle 
T = planet 



The turning sphere rotates retrogradely around an axis which is inclined to that of 
the ecliptic ; the period of revolution is one tropical year. The deferent rotates in the 
direct sense around an axis parallel to that of the turning sphere ; its period is also 
the tropical year. 

The sphere of the epicycle rotates around an axis which is inclined to that of the 
ecliptic as well as the parallel axes of the turning sphere and the deferent 98 . Its 
period of revolution is that of the synodic revolution of Mercury. 

12. Peurbach and Marcantonio Michiel 

As a last link in the chain leading from Ptolemy through the Arabs to the horoscope 
of Marcantonio Michiel, I shall now briefly discuss the theory of Mercury according 
to Peurbach's Theoricae Novae Planetarum". This work, which is indeed far superior 
to Sacrobosco's exceedingly poor composition 100 , has exercised a considerable impact 
on later Renaissance astronomy (Regiomontanus, also Copernicus, and Erasmus 
Reinhold, see footnote 84, page 122). The statement of its superiority to Sacrobosco, 

97 On account of the occurrence of two perigees, there is strictly speaking, no line of apsides as in the other cases. Cf. footnote 
62, page 107. 

88 Here, too, I refrain from discussing the theory of latitudes. It has obviously escaped the attention of Ptolemy and his 
disciples that the inclination of the epicycle (equator of the sphere of the epicycle, according to Alhazen) to the plane of the 
ecliptic must be regarded as constant. Cf. K. Kohl, I.e., p. 167, footnote 40. 

•• Cf. footnote 66, page 109. 

100 Sacrobosco's Sphaera Mundi does not deal with the theory of planets at all. 



128 The Mercury Horoscope of Marcantonio Michiel of Venice 

however, has only a relative import. Anyone familiar with the history of Islamic 
astronomy will recognize at once that there is very little in Peurbach that is 
not borrowed or directly copied from the Arabic masters, and nothing at all that 
would entitle us to speak of an independence and freedom from prejudice, such as 
is considered characteristic of the spirit of the Renaissance. 

I shall begin by giving the Latin original of Peurbach's relevant theory; the 
illustrations will be discussed below. 




Fig. 19. The spheres of Mercury (<f> ^ 0) 
P = centre of deferent (Hamil, c) 

De Mercurio 
(Georgii Purbachit, Theoricae Novae Planetarum, Nuremberg, 1472 or 73, fol. 9r.) 

Mercurius habet orbes quinque et epicyclum. Quorum extremi duo sunt eccentrici 
secundum quidem. Superficies nanque convexa supremi et concava innmi mundo concen- 
tricae sunt. Concava autem supremi et convexa innmi eccentricae mundo : sibi ipsis tamen 
concentricae. Et centrum earum tantum a centro aequantis quantum centrum aequantis 
a centro mundi distat. Et ipsum est centrum parvi circuli quern centrum deferentis ut 
videbitur describit. Vocantur autem deferentes augem aeqantis. Et moventur ad motum 
octavae sphaerae super axe zodiaci. Inter hos extremos sunt alii duo similiter difformis 
spissitudinis intra se quintum orbem scilicet epicyclum deferentem locantes. Superficies 
nanque convexa superioris et concava inferioris idem cum parvo circulo centrum habent. 
Sed concava superioris et convexa inferioris una cum utrisque superficiebus quinti orbis 
aliud centrum habent mobile : quod centrum deferentis dicitur. Hi duo orbes augem eccen- 
trici deferentes vocantur. Et moventur regulariter super centrum parvi circuli contra 
successionem signorum tali velocitate ut praecise in tempore quo linea medii motus Solis 
unam facit revolutionem et orbes isti in partem oppositam similiter unam perficiant. 

(fol. 9v.) Et fit motus iste super axe quandoque aequidistante axi zodiaci et per centrum 
parvi circuli transeunte. Motum autem horum orbium sequitur ut centrum orbis deferentis 
epicyclum circumferentiam quandam parvi circuli similiter in tanto tempore regulariter 
describat. Huius vero semidiameter est tanta quanta est distantia qua centrum aequantis a 
centro mundi distat. Unde haec circumferentia per centrum aequantis ibit. Sed orbis 
quintus epicyclum deferens intra duos secundos locatus movetur in longitudinem secundum 
successionem signorum centrum epicycli deferendo regulariter super centro aequantis. 
Quod quidem in medio est inter centrum mundi et centrum parvi circuli. Hanc tamen 
habet velocitatem ut centrum epicycli in eo tempore semel revolvatur in quo linea medii 
motus Solis unam complet revolutionem. Habet se nanque Mercurius in hoc ad Solem ut 
Venus. Fit enim semper ut medius motus Solis sit etiam medius motus horum duorum. 



Willy Hartner 129 

Ex his igitur et dictis superius manifestum est singulos sex planetas in motibus eorum 
aliquid cum Sole communicare : motumque illius quasi quoddam commune speculum et 
mensurae regulam esse motibus illorum. Huius autem orbis epicyclum deferentis motus fit 
super axe imaginario cuius extremitates sicut aparuit in Venere (fol. lOr) propter motum 
alium quern habet in latitudinem similiter accedunt ad polos zodiaci et ab eis recedunt. 
Axis tamen iste secundum se totum mobilis est secundum motum centri deferentis in circulo 
parvo. Patet itaque sicut in Luna centrum epicycli bis in mense lunari deferentes augem 
eccentrici pertransit: ita in Mercurio centrum epicycli bis in anno deferentes augem 
epicyclum deferentis peragrare. Non tamen est in auge deferentis nisi semel. Aux enim 
deferentis Mercurii non circulariter movetur circulares revolutiones complendo sicut in 
Luna contingit. Sed propter motum centri deferentis in parvo circulo nunc secundum 
successionem signorum nunc contra procedit. Habet nanque limites certos quos egredi ab 
auge aequantis recedendo non valet : sed continue sub arcu zodiaci a duabus lineis circulum 
parvum contingentibus a centro mundi ad zodiacum ductis comprehenso: ascendendo et 
descendendo volvitur atque revolvitur. Quotienscumque enim centrum epicycli fuerit in 
auge deferentis ipsum etiam motuum similitudine erit in auge aequantis et centrum deferentis 
in auge sui parvi circuli. Quare tunc centrum epicycli in maxima remotione a centro mundi 
net : et centrum deferentis in duplo plus distabit a centro aequantis quam centrum aequantis 
a centro mundi. 

(fol. lOv.) Deinde vero cum centrum deferentis per motum orbium duorum secundorum 
movebitur ab auge sui circuli versus occidentem: centrum epicycli per motum deferentis 
movebitur ab auge aequantis tantundem versus orientem. Unde centrum deferentis ad 
centrum mundi incipit accedere. Et aux deferentis ab auge aequantis versus occidentem 
recedit continue donee centrum deferentis fuerit in linea contingente circulum occidentali. 
Id autem fit cum ab auge parvi circuli quatuor signis distiterit. Et tunc similiter centrum 
epicycli ab auge aequantis versus orientem distabit quatuor signis. Aux autem deferentis 
erit in maxima sua ab aequantis auge versus occidentem remotione. Atque in hoc situ 
centrum epicycli fiet in maxima sua quam solet habere ad centrum mundi accessione. Non 
tamen tunc erit in opposito augis deferentis : nee in linea ad parvum circulum contingenter 
per centrum mundi producta. Post enim descendente centro deferentis versus centrum 
aequantis aux deferentis incipit reaccedere versus augem aequantis: centrum autem epi- 
cycli proportionaliter descendet in altera medietate versus oppositum augis aequantis. 
Unde magis removebitur a centro mundi : nee perveniet ad oppositum augis deferentis nisi 
cum ipsum fuerit in opposito augis aequantis. Id autem fiet cum centrum deferentis 
perveniet in centrum aequantis. Et tunc aux deferentis erit etiam cum auge aequantis. 
Et tarn deferens quam aequans ex quo aequales in quantitate constituuntur : erunt circulus 
unus. Et plus distabit a centro mundi centrum epicycli tunc quam distabat cum erat in 
situ ab auge aequantis per signa quatuor. Hinc autem cum centrum deferentis recedet a 
centro aequantis in suo circulo ascendendo centrum epicycli recedet ab opposito augis 
aequantis et deferentis et continue magis centro mundi propinquabit. Sed aux deferentis 
removebitur ab auge aequantis versus orientem continue donee perveniet centrum deferentis 
ad lineam contingentem circulum parvum a parte orientis. Qui punctus contactus etiam 
ab auge parvi circuli versus orientem quatuor signis distat. Tunc enim aux deferentis fiet 
in maxima remotione ab aequantis auge versus orientem. Et centrum epicycli iterum erit 
in maxima eius ad terram accessione quam habere solet : non tamen erit in opposito augis 
deferentis. Ab hoc vero loco ascendente centro deferentis versus augem parvi circuli aux 
deferentis continue revertetur ad augem aequantis. Et centrum epicycli magis elongabitur 
a centro mundi versus augem aequantis ascendendo usque dum centrum deferentis ad 
augem parvi circuli perveniet. Nam tunc aux deferentis erit cum auge aequantis: et 
centrum epicycli similiter tarn in auge deferentis quam aequantis. Unde iterum erit in 
maxima remotione a centro mundi sicut primo. Rursusque deinde similis ut iam dicta est 
mutatio redibit. 

Ex his primo videtur in anno tantum semel centrum deferentis esse idem cum centro 
aequantis. Alias autem semper deferentis centrum a centro mundi distantius esse quam 
aequantis centrum. Quare sequitur contrarium ei quod in superioribus et Venere accidit: 
ut scilicet quanto centrum epicycli vicinius augi aequantis fuerit tanto velocius : et quanto 
vicinius eius opposito tanto tardius moveatur. Secundo licet centrum epicycli tantum semel 



ISO 



The Mercury Horoscope of Maroaatonio Michiel of Venice 



in maxima ivinul iom 1 Fuetit in anno a centro uiuudi: bis tameri in maxima propinquatione 
qiiarn habere solet ipsum ease contingit, Similiter quaraquam bis in anno sit. in maxima 

.'i.eressimie : tamen Laritimi semel in anno in opposite augifi defefentis reperitiir. Torino 
ueooKHe nsf ill opposil urn atiy-is deferentis centro cpieydi extra augem aequ&ntis aut opposi- 
tion oiii.s existente (Jul, l Lr) inter centrum epicycU et oppositum augis nequantLs semper 
versrtiii-: aUquando quidern vorsu<* centrum epicyoli aliquandn ftb ad tarn praecedendo 
quam sequendn 8886 devnlveiis. Quarto sicut ,-iux deferent is ad cartas limites utrinque ab 
auge aequanfis reinovetui- ita eliam se liahet upposituni augis deferentis respeotu opposif i 
autfis aequanti.s. Maior lameii est anus huiusinodi mot us augis deferentis quam arcus 




Fiji, 2i>„ i'rnrh/rh TftsoriooB, Fol. tor: the circles of Mercury 

motus oppositi fins. I "ruin motus unius modi altorius vnloeior eril. Qumto etsi centrum 
epicycU continent ease in puncto deferentis ;» centro mnndi remutissimo mmquam tamen 
esl in punetu deferent is quctn centro mundi vicinissiminn esseeontingit. Nam dum centrum 
epieycli Itifiit in aiifjre debivntis talis est babitudo deb-rent is ut oppositum augis etus sit 
centro mundi tta vifiimm quod in quacunque alia deferentis quam habet habitudine nullus 
jmiK-iiiH eius vieinior nut lam vicinus centro mundi reperialur. In tali u Litem puncto quern 
vieiniHsimuni esse eontingil : centrum epiryeli nun . >1 <u tempore quo propinquissimuiri 
eum esse contingit; sed in cius opposite. Sexto ex dictis apparet manifests centrum epi- 
eycli jVlercuni propter motus supra dictos mm ut in alii* planriis lit: circumferential!! 
deferentis f'urulaifm sed pot ins figurae habentis similUndmetn (fob X I v) cum plana ovab 
pe rile ri.-im deseriln re. Bpicyclus veru in lun^itudinem muvetur sitait. epieyclus Veneris, 
ftevolul innein tamen tlnam in quatuor mensibus solai'ibus fere super centro BBO perbcit. 
Termini autom tabulunim hie sieid in supe riurtbus deelarantui 1 : nisi quod diversitas in 
minutis proporeionalibns aliqualis exist it. Aequatiunes enim arpiraentorum Mercurii 
quae i r i talnilis scrilninlur sunt quae eontingunt dum centrum eplcyell fuerit in mediocri 
sine b terra reraotione. I Ease atttera aocidit centro epieyeb ab au^c aoquantis per duo signa 
qual uor jj;ra.duH el XXX minuta distante. Sed in aliis planet is cent ■roepicycliin longitudinem 
media deftsi'ontis (>sistente fiebfd. Item minima ceutri epieycli Mercurii a centro mundi 
remotk) lit dum centium epicyeli »b :in^i< aequantis eins quatuor aignis distiterit. Haec 
autem in aliis oen1.ru cpicycJi in opposito augis aequantis existente eontingebat. 

Minuta igitur proporcioBalia longiora sunt exceasns remotioois centri epioyeM maximae 



Wii.i,"-, I ! un-NUH 



131 



super mediocrem eius reiiiotJoricm in srxii^mta partes acqualrH UivimiH. Hed miimla propor- 
rionalia propiora diuimtur excessus remotioms coiitri i.'pioyeli i tied ion Ls .super remotiunem 
eius jniTiiniHin similiter in LX partic-ulns jti-qiialcs dlvisus. Kt srriiintuiii hoi; duple* diver- 
sitas diametri dimniatui'. Quia tamen a \ovo inaximae aucessinnis eentri epieyrli versus 
opposition augis aequantis minuta propureicmalia pnipiom minuiiutur quae pi'his a loco 
inediocris remotionis usque ad locum maximal access ionis continue augebantui': ideo 
diqitur in Mercurio minuta propoi-ciuhalia tHplioiter ho habere: quae taintm in Venere 
atque tribus superiuribus duplieitef: in Luna vero ainiplieiter uL muni teste putuit: se 
habeas solent. 




Fig. 21. Peurbach Theoricae, KoL !l\" : the theory of tixiv; and pulcft 

Tba first of PBtrRBAGH J S figures (Fig. 20) is identical with our Fig. 17. The second 
(Fig. 21), an interesting supplementary representation of ALHAZEN'a theory, is a 
cross-section laid through the poles of the ecliptic and the apogee of the centre of the 
epicycle, Contrary to At.hazen (see above, under Section I I (c), and footnote 9S. 
page 127), Ptcukbach recognizes the necessity of making the turning sphere revolve 
in the plane of the ecliptic, and of taking care of the latitudes only by attributing an 
appropriate inclination to the plane of Mercury's epicycle. Tims by introducing the 
conception of absolute direction in space, which was not yet sufficiently developed in 
the Middle Ages, the nodes of Mercury's orbit are defined as the points of intersection 
of the plane of its epicycle with that of the ecliptic. This evidently involves a consider- 
able simplification of the numerical treatment of the problem. The plate, which 
shows the north pole of the ecliptic to the right, illustrates the moment when the 
axis of the bearing sphere (deferent), being in its apogee position, lies in the same 
plane as that of the ecliptic and of the first eccentric. Subsequently, it will describe 
the surface of a circular half-cylinder with radius ;■ above the plane of projection, 
until after one-half synodic revolution, when passing through the ptmcttwi aequans, 
it again falls in the plane of projection. 



182 



The Mercury I Icinwiipe of Muivimttmiu Mk-hirl ef 'Venice 



The third plate (Fig. 22) illustrates the motion of the apogee of the deferent (mix 
deferential , and similarly that of its perigee (called by Peukuach "oppo&Uum augis 
defwmtis"), with regard to the fixed apse -line. According to Pi-itrrach's figure 
both seem to describe discontinuous lime-shaped curves fitted between two circles, 
one of which corresponds to the deferent in its apogees position, the other to the 
deferent when its centre coincides with the punctum aequaw. Starting from <f> ■■■-■■■ C 
and $ — ISO , the two points in question will first move retrogradely on the outer 
halves of the lun.es until they reach the tangent drawn from the Earth on the small 




Fig. 22. Peurbach Theoricuc. Fol. lOr: tht> motion of the apogee of the delfemri 



circle with radius t:. Then they will move directly on the inner halves till they 
coincide with the other tangent, ; whereafter they again move retrogradely on the 
outer halves (period 1 tropical year). Fig. 23 shows that in truth the curves are not 
discontinuous. They have, of course, only one tangent also in the points of maximum 
elongation from the apse-line, and t his tangent, Q 1 t\ttQ L ' coincides with the one drawn 
from the Earth on the small circle with radius e im . Although the conception of con- 
tinuity of curves is alien to fifteenth century mathematics, it Is hardly understand- 
able that Peurbach would content himself with such an incorrect, and misleading 
figure. A discussion of the mathematical properties of these interesting curves is 
beyond our scope. 

The fourth plate (Fig. 24) shows the "oval" curve described by the centre of the 
epicycle, which we have sufficiently discussed in the preceding chapters. Again 
PjiuiUiACir's drawing evokes the erroneous impression of discontinuity for the points 
corresponding to 0" nmi </> ISO", In fact., as can easily be seen from our 

m PSflTRBACB aeemi to huvo been the ili'st to pay attention to the BBm described bv the apogee and the perigee erf tha 
df.furnnt. 
'•* Tlio two fcsBgasta (Iniwu from ttio Earth {"Qenirtm .MnnHi") <m the drete with radtoa r n.w. shown also lit Fig BO 



Willy Haetner 



133 



demonstrations, the curve has only one tangent, which is perpendicular to the apse- 
line, in each of the two points. 

I have mentioned that Peurbach's Theoricae, composed in 1460, were edited and 
printed for the first time by Regiomontanus about 1473. They enjoyed an enormous 
reputation and popularity. At least four reprints were made during the fifty sub- 
sequent years : Frankfurt an der Oder, K. Baumgardt, 1507; Basel, 1509; Vienna, 
Johann Singriener, 1518 (together with Sacrobosco's Sphaera Mundi); Basel, 1523 
(again together with Sacrobosco) 103 . There is no doubt that it was regarded as one 




Fig. 23. The curves described by the apogee and perigee of the deferent 



of the most important works on planetary astronomy in learned circles of all 
European countries. 

Thus we may safely expect also that the interior of Marcantonio's horoscope, 
which originated five years after a second edition of the Theoricae had been issued in 
Basel, was based on this work of Peurbach's. As a matter of fact, a single glance 
will put in evidence the very close relationship between the two, and therefore a few 
explanatory remarks will suffice to conclude our study. 

The designer evidently intended to represent the orbit of Mercury as seen from the 
north pole of the ecliptic, in accordance with Peurbach's first plate (Fig. 20). Indeed 
there is no other way, because only thus can the longitudes be read directly on the 
outer rim. However, his astronomical skill was obviously somewhat deficient. He 
therefore contented himself with copying the three spheres in the same situation (valid 
for the centre of the epicycle in the apogee) and approximately in the same propor- 
tions as found in Peurbach's treatise (and these proportions are wrong, like all similar 
ones extant in earlier works). Then he placed the epicycle carrying the disk of Mercury 
in such a way that the straight line joining the Centre of the Universe (Earth) with 
that of the disk of Mercury indicates the "apparent" longitude of the planet. The 

103 Cf. Zinner, Oeschichte und Bibliographie der astronomischen Literatur in Deutschland zur Zeit der Renaissance, Leipzig, 
1941: Nos. 890, 918, 1098, and 1216. 



I'M 



Tlio Mcivury Horoscope of Mareantonio Michicl of VenJoe 



situation of the planet in the epicycle is also indicated schematically, because, at 
the moment in question, Mercury did not occupy the perigee of the epicycle. More- 
over, the designer committed the error of placing the small circle with radius e in the 
centre of the whole disk, with the Karth as its centre, whereas he ought to have 
placed it eccentrically. Apart from this obvious lapse the designer could not possibly 
arrive at a satisfactory solution of his problem except by drawing the figure in correct 
proportions, as is done in our Fig. 26. 

There is one more very characteristic symptom of the designer's insufficient 
training. The three parallel broken straight lines perpendicular to the apse-line (this 
latter is not drawn in Makcantoxio's figure) make no sense here. We do not find 




Ki^. 24. Pwrbaeh Thvoricae, Fol. I Lr: the oval mrvc of the centre of the rpii-ycle 

them in Prukhach's first plate, but they play an important part, in the sccimd (Fig, 
21). being the tens of the spheres ! hi other words, our designer has here confused the 
projection of the ecliptic with the cross-section in the plane of the circle of longitudes 
through the apogee. 

The last question concerns the longitude of the apogee itself, which in the figure 
of the horoscope is ca. Libra 29° == 209°. We do not know whether this figure was 
really computed for the year l;727 or directly taken over without eomputation from 
an earlier source. It is almost exactly 20° larger than Ptolemy's figure (which refers 
to the epoch ca. I :J4 a.d. 1(H ). As the interval between the two is nearly 1400 years, this 
wotdd indicate a precession constant of I in ea, 70 years, which would correspond to 
the best value known in the Middle Ages: riz. t he one determined, about I2fin A.n. f 
by Nasik al-DIn al-TOsI. A comparison with other values of the longitude of the 

"" Sm footnote Ok lpjitfi- icm. PTOUTOT'S euul fliiiiiv, lor IM A.D., is* Mhni !I 1.V I*'l 15*. The iltHViriu* of 8 , which 
X.il.i.i.vo (al-KattSnIj f>/». Axtf.. I, p. i\i) riTiird« &a uiu*x]il:i1in-il, nppmxlmuti;lv i-orrrH|ioJii|H Id Oie time t'liipwii lii'twcon the 

era of ETABOHi&BAB unci tluit of 1'tulkmv, wliirh Is «a. 'hid ywm. X.o.us iBtakenb l-.-fir\.., ■ t •■ ■ iliiitr.- lsl 10', whldi is 

PTOLBKY'e value rnriipulcii Ibr tin.- llr*i ye&l • n|' N.miiiyissak. Id refer In I'Ttii.KMV's tivin iK-rlixJ. 



Willy Hartner 



135 



apogee that I know of yields less satisfactory results : 875 a.d., al-Battani : 201° 28' 
(l°in87 a ); 1029, al-Biruni: 203° 43' (l°in 94£»); 1204, al-Jaghmini: 206° 23' 33" 
(1° in 127a). 

If however Marcantonio's astronomer really had employed Nasir al-Din's 
value of the precession, it would be inconceivable that he did not also employ his 
value of the longitude of Mercury's apogee. I have been unable to find the latter, but 
it must evidently be ca. 48' larger than al-Jaghmini's, viz. ca. 207° 10'. To the re- 
maining 1° 50' would correspond 130 years, which would bring us to the year 1390, 




Fig. 25. Interior of Marcantonio's horoscope in correct proportions 

Longitude of apogee, A = 209° 

Mean anomaly (/_AE'H), <f> = 244° 

"App." anomaly ( /_AEH), v = 247° 

(Mean) anomaly of planet in epicycle (/_gHT), t = 187° 

"App." longitude of Mercury ( /_ °pET), L = 92° 

Dotted circle (centre E) = circle indicating mean geocentric distance 

whereas for the year 1527 we would have to expect a longitude of 207° 10' + 3° 48' 
= ca. 211°, or 2° more than indicated by the horoscope. 

The exactness of Marcantonio's figure is not sufficient, and the discrepancy not 
large enough, to allow of a reliable conclusion on this point. 

13. Concluding Notes 

Thanks to the generous co-operation of the Director of the Bayrische Staatsbibliothek 
and the good offices of the Central Administration of the Frankfurt Libraries, one 



186 



Tlir HSeaJOury Horo&eopo of Maroanfcoriio Michiet of Vonice 



of the extremely rare copies <>!' -Inn a\m;s Nt-owi. wit's Almanack nova phtrimis annis 
vmiwix inservimf.w w ' containing planetary ephemerides for the years 1499-1531, 
was placed at the author's disposal just in tame bo make the following additions and 
corrections to the present article. 






~ TZ r~ '.*it« , ,iBalaR.C,E a|MCi 
j £ I _ ^_J f»_4_ tt i <P i fit'S i A 

i a i «■ v i ii i «n i ep S * s 

li "'.-1 . • |J • lS "li '"i|i *|J *|j m 

■ li»_ irl.f l t l t l,tl» Hl|:r i-l J i-||.' J, |(4 f, 
Iji a ur y - ii-.jh l+;jy> C.Ji J,.|'. 4|n '4144.' 
;;'■ mj »" tvii> i ■■■=•■•■ i M Mil 'Mi-. HIH 44 

~fl« f| 1 ' I.. :, . .. . , , , | 4 t|, ^|-. 4 ,< 

- d |"4 *)if i*ji» w iM o d!ij 44'" Ifjii i . 

fl<4} ft**;. H.\ o° 1^Za53V~*" Wl 

H|!r f*]n — Bj o luj I .NJ4 <?' « ll 

H r i • I r ■ ■ ■ • , ^ ; , ■ , , 

■W <■' J ' l> f "I I *^H »V» 411 > »fN" iJ 

-.Jl* 4'lii .IK » IV H . . k 11! 4 i,|i. 



, Hj*3tJ"fc , *l"! '*': ' "I" '+'4 "I 4 4 

Ml ' _S1 4*il| a 41 1 _,»_»»!" iSj'f I'll 4MH_? 

I :-. -I I 

I 1**14 I 



^ 



(I V;,, III ' 4»| i '.I.I 

*Li_M "■ " - ■ ' ' ' a' 1 

• 1 4 'Tji* u "I * " 

. <%*«_J«l_f„'«L*._ '< ! ■" '•. ' ''■<-•■• 
<wrt ■*! «_u|t» v 4-i • 

lei > iKi'llI i.) 4 > : 
_**•*!'. '• I * ~i*i] ii ~ 

E. * 

bip.J4 



.in • 

4"l 1/ 
Ii .. .< 



, l.< . 1.1 4. !..!. 



I 4«|H fl , |i| r > 



t-L I 



IP *W 4 4 



5**«I» 



11 -V ...n tc i ■ HI 4 4i;it Ii! 4 t||l« 4III4 41 
14 >< V.-i'^, .'- ■ 'IS 1 4 <*U ),. . II 11 ... 

"I" _. l f * ** ' "I f "*" .''I s *5f* M «|H '■ 



"I" 

14 •■ 



** < ■ 1 1 '"y-i^L' J -'I ' _ '4|' 1 1. 1 » i»)i< •" nju ] ■ 
win . nt.?^ »* » * 1 1 ~itu m m iiin' i-jM if 

UW Btj 1 1 HJ m I 4lL 1 lj lit 4tf w «B4 trln * l« 



Iwb plmjjn™ id wm 



I MJ ^ lij I 4.1| a »JJ B (M\ 

j! 1 _!•) s .'I : .41 . 1^ 4" flj 



t^«lto< J fiUnr A i^l.^.T '« 




- 1° <l j Q - 

1 Id h 

1.1 1 * .i(a ill _ ! 




Pig, I'll. 'I'wii jitiiir-: n-iuii Stm-kklkh's Alwitnttch HotXt; KpliiMiu-nVlrs for June, 1-527 

(«) Under the date 15th June, 1527, the planetary positions listed on the left-hand 
page 1 (of. Pig. 2fi T showing the two pages for June. tr>27) are as follows: 



Sim Moon iSiLtitm Jupiter Bfara 

Cue 2° 33' i Cap 10° 33' ■ Tm <« MV \ Cue 2° 53' i Sco IT 2T 



Vomim B fo r enry 

One 20 r,' cue 2° 33' 



Ascending 

Node 

S{jt 24 Q G' 



Iii comparing these with the positions indicated on the horoscope: 



Sun Moon 

I'm- ■> Cap 20 



Saturn 

T,m I 



Jupiter 
One 2 



Man Venus 

Sea 28" Cm 23 



Meri'tii'y 

Cn© sr* 



ABoending 

Node 

Sgr 24 p 



we hoo that I he latter were directly taken from the former, not by interpolation but 
by rounding off in a very arbitrary and unsystematic way. As the positions given by 
SrBtfFLBB refer to mean noon on the 15th June, while the horoscope was cast for 
8 a.m. nl'thu same day, the correctly interpolated values would have been as follows: 



■Sim Mi tint Sulnrn 

Cur 2 J '!»' Cap 17" (i' Tnu 0" I.V 



.lupitm- 
One 2' Bl* 



Ase finding 
NckLo 



Miirs VettOS !Mon:iiry 

SCO -':! l'-r One W 3 :W Cnc 2 3 47' Sgr 24° G' 



""'i I'clntol by .Ion annus RBQBb, Him, 1 100, MO Secttoa &. 



Willy Hartnkr 



137 



jiih! when rounded oft" correctly (prescriptions for this procedure arc ^ivon in 
Stoffleb's introduction), as evident, 



Bud 

Quo 2 



.M [ M > 1 1 .Silt Hill -Jll|}iU't > MlLl'tt VotUtS 

Cap 17" Tim t 1 Oem 3" SaoOT One 27* 



Mercury Ni(i|( . 



Ono 8* 






It seems impossible to decide whether the incorrect figures of the horoscope are 
due to the astrologer's lack of mathematical insight alone, or whether he wanted the 



_ i »ti iv i it i m i = i « i > 

lhpdtW flJ I ■., jII.h H|,| te lj t\M 111 k n| I II 
fcl^OM ll| m-l til>. <!■• ill? «*l'l *1'^ *l ' V 



'■■-■:. ^r 



hi i M .v n .,>.. i|» ti'l? 4«)n 8(" *l i 9 

i!i > ■!•<• w)in »,rt »ij is +r >■+ 'ij^ "i >_* 

irn ...i» y », i'i niii" iMii i4't W" J'l'. l 

~lj»« Mf l'l" <HW up*' Hjt » <-1i» HM-' "I. [» 



?im i jjfli-jsi't ill" iy» ..*'1>*J4?_ | ti_n 



WH^- fll i Jl h|ii n >j ii ** i: 

BpjlWMM. t J » n I3| 14 _a|.f 4^11 JIIID (I'll 1 71 » ll|'^ 



g 'j <twMhafc-.il 



wfc-.il ■ iB ii. wt tf fR» W 1 if ff ■ ■Ti_N a M 
nil wi'>^/7lir m« «■» 48w«~i a} T a f»l ■> ■¥» 

ul t l»1itrt~ Ht* «« l4" <H_'4l l ?_*LiJ' 

UN 1\* *>!■< * "In <t|» -H|i«'»i« ■*T*' H 




m* ill 



«fi»_ [«|j 9 1. 1 a <i 1 

jl4_M_l __ 

^"|i|if~*1i a >f 

,»j •-4' I « 4«lJ»_!l?__ti 

•4 T >||S Jl S 






|il ~"l t|iT Wliil »|iS 4?) if * «|M* nli«_4« 

MM ,i]-l «\ » mI . "Jo lift KtfJ*. 



aatofriln^itiw ~»T> "1 «( ' 


Ml^^f, 




i=| 1 St » Up 


fT-^i 









(17 ftiptxM* hint fclielrtm^ii w 1 



"5a»«(limijn*>fc(B(i," 



*,V I O I It in i iT I V .? 1 

I « h I tr if Ml 

■ — 1 — i T' — L^_L_ -- 




1 I 1 '^ Hlftam 

; i 1 TTT 'u^® 



1Q<TB 






j I (0 !lL 



011-9 



a 



i 



Fig. 27. Two pugc» fVom Stokklkk : Ephcmcrides for hVhrmuy, 



double eon] unction to take place in Cnc 2° instead of 3°, which would have been 
more correct, J can sec no astrological reason for it, nor do I see any for neglecting 
the Moon's motion of ea, 2 ? ;"K or for making Venus stand in Cnc 2(i* instead of 27°. 

\h) From t ho circumstance that Stoffler's Almanack contains tables for the com- 
putation of the twelve astrological Jwases for the latitudes 42 -54°, which are based 
on Regiomontanus', not Ptolemy's, system, we may conclude that also the homes 
figuring in Maroantonjo Mtciitkt/s horoscope were based 011 Reuio.moxtamus 106 . 
The distribution of the hotwes, fines, and fades then is as shown in table overleaf. 

In comparing this (able with the ones given in Sections ~> {h) and (» («), (h) w } we find 
that three out of the twelve cusps (Houses No. Ill, IV, and VI) fall on onpropitious 
fines, against one according to Ptolk.mys system, and four (Xos. IV, VIII. IX, XI) 
on unpropit Sous fades, as against three according to Ptolemy. The general character 
of the horoscope, however, is not altered thereby, especially because the four main 
ftouses (I, IV, VII, X) are the same in all existing systems. 

■" HKiii<iMiivrANrs*nbi]!viiU'Hini'li of the (burcf|unl divisions oT ttw n/mitor vhivh niv nuiliiuil hi-twii-n lln-1nirlwiu;uiii ilio 
meridian into three equal f«ccMtiii!i or 80 . Tin- totuw. then, are the unequal «eetlt>n» ul" the oollpttc oil out to the "CtrcJaa or 
liosLUnii" (Willi jiulia hi Lhu iiorlh and south uolntfl pf the liurlMin) laid throiiKh the twelve cijual m:clionH o|" (lit; »-tni:itur 

" : Sit 111 imrticuliir the miond lahle on p. lot), 



138 



The Mercury Horoscope of Marcantonio Michiel of Venice 





Finis 


Fades 


House No. I . 


. Leo 


18° 


IV, Mercury 


II, Jupiter 


II . 


. Virgo 


9 


II, Venus 


I, Sun 


Ill . 


. Libra 


3 


I, Saturn 


I, Moon 


IV . 


. Scorpio 


6 


I, Mars 


I, Mars 


V . 


. Sagittarius 


16 


II, Venus 


II, Moon 


VI . 


. Capricornus 


22 


IV, Saturn 


III, Sun 


VII . 


. Aquarius 


18 


III, Jupiter 


II, Mercury 


VIII . 


. Pisces 


9 


I, Venus 


I, Saturn 


IX . 


. Aries 


3 


I, Jupiter 


I, Mars 


X . 


. Taurus 


6 


I, Venus 


I, Mercury 


XI . 


. Gemini 


16 


III, Venus 


II, Mars 


XII . 


. Cancer 


22 


IV, Jupiter 


III, Moon 



(c) According to Stoffler's ephemerides for February, 1527 (Fig. 27), the 
conjunction of Mercury with the Sun did not occur on 7th February as computed by 
me (see Section 8), but on 14th February, and the one with Venus, not on 11th, but 
on 17th February (see Stoffler's last column). On 14th February, the Moon is 
listed as standing in trigonal aspect with Saturn at 6 p.m. (col. 3) but simultaneously 
approaching quadrature with Mars (occurring 20 hours after noon, on 15th February, 
at 8 a.m.); therefore, this day could not be found appropriate. On 17th February, 
however, we find that the Moon stood in the propitious sextile aspect with Mars 
(at 9 p.m., see col. 5) and that no unpropitious aspect disturbed the marriage of 
Marcantonio 's celestial protector. Hence there can hardly be a doubt that Marc- 
antonio himself celebrated his own marriage with Maria Soranzo on the same day. 

(d) From a comparison of Stoffler's ephemerides for 7th and 11th February 
we see that the deviations of the planets from the true positions, as calculated from 
NT II and III, are very considerable. In the case of Mercury, the error amounts to 
no less than 12°; for Venus, Jupiter, and Saturn the positions are given fairly cor- 
rectly, as could be expected, but Mars again deviates ca. 6°. This shows, on the one 
hand, that there are problems of dating, like the one treated in Section 8, that 
cannot be solved with the aid of modern computation but only by resorting to the 
tables or ephemerides actually used at the time in question ; and, on the other hand, 
that the incorrectness of the tables must have been felt so disturbing that a complete 
revision of the foundations of Ptolemaic astronomy seemed inevitable. This revision, 
as will be remembered, was almost completed by the time Marcantonio 's horoscope 
was cast. Only sixteen years later, the scientific world witnessed the dawn of the 
new era of astronomy. 



Observatories and Instrument Makers in the Eighteenth Century 



D. W. Dewhirst 

The Observatories, Cambridge 



1. During the last decade a renewed interest has been taken in the study of early 
scientific instruments, not only for their intrinsic interest as examples of the crafts- 
man's art, but also for the light they throw on the instrumental limitations imposed 
on the ability of the experimental scientist. The exacting requirements of positional 
astronomy have always made greater demand on the skill of the instrument maker 
than almost any other science, and this interplay between the observer's require- 
ments and the ability of the instrument maker to satisfy them is especially significant 
in the rapid development of this branch of astronomy during the eighteenth century. 
The topic is here considered with reference to Bernoulli's account of the instru- 
mental equipment of the various observatories in Europe in the latter half of the 
eighteenth century. 

2. The Jean Bernoulli here referred to was Jean Bernoulli III, born at Basle in 
1744. He was the eldest son of Jean Bernoulli II (1710-1790, professor of 
mathematics at Basle) and a grandson of the celebrated Jean Bernoulli I (1667- 
1748). He inherited the distinguished mental ability of this remarkable family, 
took his doctor's degree in philosophy at the age of thirteen, and was called to Berlin 
as astronomer by the Berlin Academy when only nineteen. During October, 1768, 
to July, 1769, he undertook an extensive tour of the observatories of Europe and 
wrote an account of what he had seen (Bernoulli, 1771). This account is 
published in the form of letters to a hypothetical correspondent, which (as he 
explains in the preface) enables him to give, by means of the dates of the letters, the 
exact time at which he found things as he describes them, and also to beg his readers' 
indulgence for the style of writing. This latter is very informal, and the letters con- 
tain impressions and personal anecdotes expressed in a frank simplicity which makes 
them not only entertaining to read, but sometimes a valuable source of otherwise 
inaccessible background material. The Lettres Astronomiques are of greater impor- 
tance, however, in that they describe the interest and reactions of a young but 
knowledgeable astronomer on visiting the principal observatories of Europe in his 
day. Although known to such bibliographers as Houzeau and Lancaster the book 
does not seem to have received much attention from previous historians of the 
period, perhaps because copies are now rather scarce. 

3. The revival of observational astronomy in Europe in the sixteenth century took 
place largely in observatories established by the zeal of individuals and financed 
either from private sources or by noble patronage. Tycho Brahe and Hevelius 
both expended private fortunes on their observatories, whilst the Landgraf of Hessen- 
Cassel was not only an able observer but a munificient patron of the science. The 
foundation of such bodies as the "Invisible College" in England in the 1640's, which 

139 



140 Observatories and instrument makers in the eighteenth century 

became the Royal Society in 1662, and the Academy of Sciences in Paris, gave 
great impetus to the experimental sciences, both by providing a medium for the 
exchange of information and the diffusion of learning throughout Europe, and in 
part by providing further encouragement and patronage for the individual of ability. 

4. During the latter part of the seventeenth and in the eighteenth century many of 
the universities also established observatories which were used both for instruction 
and research. These also, however, usually owed their existence to the zeal of 
individuals rather than to any determination to prosecute astronomy as an integral 
part of the work of the university, and were frequently little more than an adornment 
to the apparatus philosophicus of the institution. Professor Stratton (1949) has 
described the early history of observational astronomy in Cambridge, which well 
exemplifies these points. During the hundred years from 1703 there were three 
separate observatories in Cambridge which flourished each but a few years, whilst 
many private individuals had collections of instruments in the several colleges : no 
original contributions of great moment came from any of them, although they were 
occasionally used for elementary instruction. Price (1952) has given a detailed 
description of the instruments of the first of these observatories, namely that at 
Trinity College, many of which are still extant. The instruments acquired in 1703 
included a Gunter's quadrant of 13 in. radius "with a nocturnal on ye backside", a 
telescopic level, a universal ring dial of 10 in. diameter, and a 16 ft telescope with 
two object glasses. There were numerous minor items, "a plain table with its furni- 
ture" (used in surveying), "a concave burning metal", "Mr. Molyneux his telescopic 
dial", etc. It is evident that an institution with such equipment, although referred 
to by a contemporary as "a stately astronomical observatory, well stor'd with the 
best instruments in Europe" was not really equipped to prosecute valuable research 
even if a constant supply of able observers had been forthcoming. In fact the 
observatory fell into disuse within a few years of its completion, and although the 
building existed until 1797 Bernoulli did not regard it as worth a visit when he 
was in Cambridge in 1769. 

Whilst taken only as an example, this history typifies many of the university 
observatories described by Bernoulli later in the century. In England, it is true, 
the tide of intellectual activity in the universities was perhaps at its lowest ebb in the 
mid-eighteenth century, while in the mathematical sciences in particular the 
development of Newton's work was stifled by the continued use of the master's 
fluxional calculus, and by a rigid adherence to methods of investigation which 
Newton had virtually exhausted. This is perhaps a contributing factor to the lack of 
inspiration in observational astronomy at least in the English universities of the time. 
There were, of course, a few notable exceptions, as exemplified by Tobias Mayer, 
who became director of the Gottingen Observatory in 1751 and carried out, with a 
mural quadrant of 6 ft radius constructed by Bird, a notable series of accurate 
observations for use in the construction of his solar and lunar tables. We may also 
note a series of accurate observations made by Hornsby in Oxford with a similar 
instrument some years later. 1 But these exceptions resulted usually from the 
combination of an individual of ability and a good instrument; if nothing else, 
the changing population and uncertain support of university observatories were in 



1 See Dr. Knox-Shaw'S contribution to this volume, p. 144. 



D. W. Dbwhibst 141 

general not conducive to the sustained work of high quality necessary for good results 
in positional astronomy. 

5. We may consider briefly the sources of the instruments used by observers prior 
to and during the eighteenth century. The main centres of instrument making in 
the sixteenth century had been in Germany (especially in Augsburg and Niirnberg), 
the Low Countries, and in London. The principal products were small portable 
dials, surveying and navigational instruments, of high artistic merit and skilled 
craftsmanship. When Brahe first started to use the divided instruments of large 
dimensions which characterized later work in positional astronomy, he at first sought 
the assistance of the Augsburg instrument makers — as exemplified by the great 
Augsburg quadrant (Brahe, 1598). But in general the specialised skill of the instru- 
ment makers were not immediately adaptable to the astronomer's needs, and most 
of the great observers of the early seventeenth century relied on their own skill to 
produce mountings and divided circles. At this time, too, the larger optical com- 
ponents were made by the observers themselves: when James Gregory took his 
plans for the reflecting telescope, which he had described in Optica Promota (1663), to 
London, he was unable to find an optician to polish the necessary surfaces. Both 
Newton and Hooke made the first reflecting telescopes with their own hands 
(Birch, 1756). 

6. The growth of the Royal Society in London and similar academies in Europe 
at about the same time had two further influences on the development of observa- 
tional astronomy. On the one hand, the increased demand for philosophical apparatus 
stimulated the clock makers and spectacle makers of the capital cities to take an 
increased interest in the demand of new clients. Hooke's diary (1672-80) contains 
frequent references to his visits to such famous clock makers as Thomas Tompion 
(1639-1713) and opticians like Christopher Cock (sometimes Cocks, Cox, etc.), 
fl. ca. 1660-1680, who was, indeed, referred to by Oldenburg, Secretary of the 
Royal Society, as "our perspective maker Coxe . . .". Many of these craftsmen 
were men of considerable ability outside the requirements of their profession, and 
like George Graham (1673-1751), the clockmaker, were later to become Fellows of 
the Royal Society in their own right. 

On the other hand, influential support could be given to a new type of observatory : 
the national observatory, supported by State funds for the prosecution of observa- 
tions generally recognized to be urgently needed. The Royal Observatory of Paris 
was completed in 1671; in England a number of Fellows of the Royal Society, 
considering the long standing difficulty of the determination of longitude at sea, 
suggested the construction of a small observatory where observations could be directed 
to this problem, and John Flamsteed commenced his duties as the "Astronomical 
Observator" in the new Observatory at Greenwich in 1675. Despite the smallness 
both of the building and of Flamsteed's salary, the observatory was better designed 
and regulated than that of Paris, which had an unfortunate history in its early years. 

At first, Flamsteed sought through Hooke the assistance of the clockmaker 
Tompion in the construction of his apparatus, but he eventually employed the 
talented mathematician and then amateur instrument maker Abraham Sharp to 
produce a mural arc of 140° and 79 in. radius which was undoubtedly the most accurate 
divided arc ever produced up to that time. This was the beginning of a distinguished 



142 



Observatories and instrument makers in the eighteenth century 



partnership at Greenwich between instrument maker and astronomer, the main 
course of which may be outlined as follows (Table I). 



Table I 



Astronomer Royal 


Instrument 


Maker 


Notable Observations 


John Flamsteed 


50 in. mural quad- 




Obliquity of ecliptic, etc. 


(1676-1719) 


rant 1683 




Fundamental right ascen- 




79 in. mural arc 1689 


Abraham Sharp 


sions. 
Catalogue of the "Historia 
Coelestis", 


Edmund Halley 


66 in. focus transit 


Robert Hooke ? 


Zodiacal stars. 


(1720-1742) 


instrument 1721 




Places of Moon for determi- 




8 ft mural quadrant 


George Graham 


nation of longitude. 




1725 






James Bradley. 


1721 transit repaired 


Jonathan Sisson 


Latitude and place of equi- 


(1742-1762) 


1725 quadrant modi- 


George Graham 


nox. 




fied 




Refraction. 




12 ft zenith sector 


George Graham 


First general transit obser- 




1727 




vations of high order. 




8 ft radius mural 


John Bird 


(Aberration) and Nutation. 




quadrant 1750 








8 ft focus transit 


John Bird 






1750 






Nathaniel Bliss 


— . 


— 


Continued Bradley's work. 


(1762-1764) 








Neville Maskelyne . 


Commenced work on 


. — - 


Fundamental stars. 


(1765-1810) 


new mural circle 




Places of Sun, Moon, and 

planets. 
Nautical Almanac from 1767. 


John Pond 


Mural circle 1812 


Edward Troughton 


Catalogue of 1112 stars 


(1811-1835) 


10 ft focus transit 
1816 


Edward Troughton 


(1833). 



The successive instruments by Sharp, Graham, Sisson, Bird, and Troughton 
may each be said to mark an epoch in the art of instrument design and circle dividing, 
as the accuracy of the Greenwich observations through this period bear witness. 
Missing from the list are Jesse Ramsden, F.R.S., fl. c. 1762-1800, whose greatest 
contribution was the dividing engine, and, of course, such opticians as Dollonds 
(1750 onwards) and James Short, fl. c. 1735-1768, who was the first person to make 
successful reflecting telescopes commercially, and whose labours amassed him a 
fortune of £20,000. 



7. Space has not permitted reference to the contributions of Ole Romer (1644- 
1710) and Johann Tobias Mayer (1723-1762) to the construction and design of 
divided instruments, and there were, of course, developments in instrument manu- 
facture on the continent allied to those in England in the eighteenth century. But it 
is clear that the continental instruments makers did not vie with those of London in 
the provision of the capital instruments of the world's observatories until the 
nineteenth century. 

That this view is not a result of faulty historical perspective or prejudice is con- 
firmed when we revert to Bernoulli's Lettres. A Swiss astronomer, interested in 
instruments, he notes in almost all the cities he visits instruments by Sisson, Bird, 
Dollond, and Short, and especially regards the divided instruments as the best 
available in his time. After visiting Greenwich for three days, he wrote on 26th March, 
1769, "Je finis ici cette lettre; elle est bien longue, mais c'est aussi du principal 



D. W. Dewhirst 143 

Observatoire d l'Europe que je vous ai parle, et je crains bien de n'avoir pas traite 
le sujet comme il meritoit" (p. 99). But despite his enthusiasm for what he saw at 
Greenwich, his first concern on arriving in England was to visit not observatories 
but the instrument workshops : he saw the "very considerable" one of the Dollonds 
in St. Paul's Churchyard, most of the shops of the numerous philosophical instrument 
makers, and wrote later on 3rd May, 1769, about the instruments of Bird, Sisson, and 
Ramsden. The precision of the divided circles of these artists he places in that order, 
"dans le quels ils sont mis par le plus grand nombre de ceux qui connoissent leurs 
ouvrages". Bird's workshop made instruments of all types, including the necessary 
lenses (unlike some of the other makers, who relied on other firms for their lenses), 
and Bernoulli notes that his divided instruments were the most expensive: a 
5 ft mural quadrant cost 250 guineas. Of Ramsden, unfortunately, he does not tell us 
much, but Ramsden had completed his first dividing engine only three years pre- 
viously, and was yet to achieve the fame that enabled him to equip the observatories 
of Palermo, Florence, Padua, Mannheim, and St. Petersburg — to name a few — with 
their well-known transit instruments and vertical circles. 

Apart from the individual genius of these artists and the stimulus of the astro- 
nomers' needs, there are perhaps contributing factors to their success : some were 
related by marriage and shared technical information within the family, whilst 
others had been trained as apprentices by their predecessors : always there was the 
inducement of the rewards offered by the Board of Longitude for such improvements 
in instruments, clocks, or the art of circle dividing as might lead to the ready deter- 
mination of longitude at sea. But whatever the cause, the observatories of Europe 
clamoured for their productions, and frequently the makers had difficulty in meeting 
sufficiently quickly the orders placed with them. 

Bernoulli concludes his first letter from London: "Me voici a Londres depuis 
8 jours; je ne puis vous parler encore d'Astronomes ni d'Observatoires; mais je 
vous ferai part de la surprise agreable ou est jette un Astronome en parcourant les 
rues de cette Capitale. Vous aves surement oui parler de la richesse et de l'eclat des 
boutiques de Londres, mais je doute que vous vous representies combien l'Astronomie 
contribue a la beaute du spectacle: Londres a un grand nombre d'Opticiens; les 
Magasins de ces artistes sont remplis de Telescopes, de Lunettes, d'Octans, &c. 
Tous ces instruments, ranges et tenus proprement, flattent l'oeil autant qu'ils 
imposent par les reflexions auxquelles ils donnent lieu". (8th December, 1768.) 



References 

Bernoulli, Jean 1771 Lettres Astronomiques, oil Von donne une idee de 

Vetat actual de I Astronomie pratique dans 
plusieurs villes de VEurope. Berlin. Chez 
l'auteur. (8vo., 175 pp. and 2 plates.) 

Birch, Thomas 1756 History of the Royal Society, London, vol. Ill, 

p. 122. 

Beahe, Tycho 1598 Astronomiae Instauratae Mechanica. (English 

translation by H. Raedee, E. Stromgren 
and B. Stromgren, Copenhagen, 1946, p. 89.) 

Hooke, Robert 1680 Diary of Robert Hooke, 1672-1680. Edited by 

H. W. Robinson and W. Adams; London, 
1935 (passim). 

Price, D. J 1952 Annals of Science (London), vol. 8, p. 1. 

Stratton, F. J. M 1949 "The History of the Cambridge Observatories", 

Annals of the Solar Physics Obs., vol. I. 



The Radcliffe Observatory 

H. Knox-Shaw 

Formerly Radcliffe Observer, Pretoria, South Africa 1 



As Professor Stratton has been for several years one of the Radcliffe Trustees, it is 
perhaps not inappropriate to include in this volume a short history of the Radcliffe 
Observatory. 

When Dr. John Radcliffe, who had been London's leading physician during the 
reign of Queen Anne, died in 1714 he left a considerable fortune. His will directed 
that the residue of his estate, after several specific bequests, should be employed at 
the discretion of the trustees for charitable purposes. Now in English law any 
institution the sole purpose of which is the pursuit of knowledge is classed as a charity. 
So in 1771 the Radcliffe Trustees agreed, on the application of the University of 
Oxford, to build an observatory for the use of the Savilian Professor of Astronomy. 
A magnificent building on nine acres of land on the then confines of the city of 
Oxford was erected in the course of the next twenty years, the architects being 
Henry Keene, who had been responsible for the Radcliffe Infirmary, and James 
Wyatt. In addition to library, lecture room, and accommodation for the astro- 
nomical instruments there was a commodious house for the director of the Observa- 
tory, who became known as the Radcliffe Observer. The Observatory was thenceforth 
entirely controlled and financed by the Radcliffe Trust, and until nearly 160 years 
later, when the question of moving the observatory to the southern hemisphere was 
being considered, there were no astronomers among the Trustees. Since then, how- 
ever, the Trust has had the advantage of the technical advice of Sir Frank Dyson, 
Sir Arthur Eddington, Sir Harold Spencer Jones, and Professor F. J. M. Stratton. 

The Rev. Thomas Hornsby, Savilian Professor of Astronomy since 1763, to whose 
energetic action the founding of the Observatory was due, had at his disposal an 8-ft 
transit, two 8-ft mural quadrants, and a 12-ft zenith sector, which were the last and 
probably the best instruments made by that famous craftsman John Bird ; and 
very good use Hornsby made of them, for he observed the positions of the Sun, 
Moon, planets, and brighter stars assiduously and almost continuously for nearly 
thirty years until stopped by ill health in 1803, when he was seventy years of age. 
He made all these observations, some 80,000 transits and 20,000 zenith distances, 
himself for he had no assistant and, as a reduction of them made 120 years after 
Hornsby's death has shown, they were of a high order of accuracy for those days. 
It is astonishing that one man should have been able to observe a star in both 
co-ordinates at one transit, as he frequently did, taking the time (by the eye-and-ear 
method, of course) over the first three wires of the transit, then going to the next 
room to observe the star's zenith distance and returning to the transit in time for 
the last three wires. This remarkable man also held the posts of Radcliffe Librarian 
and Sedleian Professor of Natural Philosophy for most of the time he was observing ; 
and he managed to produce the first volume of Bradley's observations. It is thus 

1 Now at Elgin, C. P., South Africa. 

144 



H. Knox-Shaw 145 

not surprising that he had to leave to other hands the publication of his own 
observational work. 

When Hornsby died in 1810 he was succeeded by the Rev. Abram Robertson, 
who continued to observe with the same instruments, though with the aid of an 
assistant, one of them taking the transit and the other the quadrant. The next 
Observer and Savilian Professor was Stephen Peter Rigaud, who was appointed in 
1827. He observed very little himself, as his interests lay in the direction of the 
literature and history of science. He amassed a large library of early scientific 
works, which on his death was purchased by the Trustees. Three years before his 
death in 1839 a 6-ft meridian circle by Thomas Jones was erected to take the place 
of the two quadrants. In design it was transitional between the mural circle and the 
meridian circle proper and seems to have been unique. 

These first three Radcliffe Observers held also the Savilian Professorship of 
Astronomy, and the appointments to the two posts were made by agreement between 
the Radcliffe Trustees and the Board of Electors to the Savilian Chair. However, in 
1840 these two bodies failed to agree on a successor to Rigaud. G. H. Sacheverell 
Johnson was elected to the Professorship, but as his subsequent career showed (he 
exchanged soon after to the Whyte Professorship of Moral Philosophy and later 
became Dean of Wells) he had no particular interest in astronomy. The Trustees 
were naturally anxious to have someone with experience of astronomical instruments 
and observation as their Observer and appointed Manuel Johnson, who as an officer 
in the St. Helena Artillery had been in charge of the observatory on that island and 
had compiled a catalogue of six hundred southern stars. From this time onwards 
there was no direct link between the University and the Observatory, the use of which 
was lost to the Savilian Professor. On the other hand, the Observer, freed from the 
duty of teaching, could give his whole attention to the needs of the Observatory. 

Manuel Johnson thoroughly justified his selection, for he proved himself an 
energetic and resourceful director. In the nineteen years before he died at the age of 
fifty-four he introduced several new activities into the Observatory's routine. 
Amongst these was the publication of an annual volume of Badcliffe Observations 
containing astronomical, and a few years later also meteorological, results. The 
transit was remodelled by William Simms in 1842 and was in constant use along 
with the Jones meridian circle. In 1848 a 7|-in. heliometer by Repsold, the only 
example of an instrument of this kind ever erected in England, was installed in a 
separate dome. It was used by Johnson and his assistants for measuring double 
stars and the positions of minor planets. A second assistant was added to the staff 
in 1851. He was Norman Pogson, the originator of the proposal that the "light- 
ratio" defining the scale of stellar magnitudes should be the fifth root of one hundred. 
The devoted services as a computer of William Luff, given for many years gratui- 
tously, commenced in 1844 and were to last until 1889. The longitude of the Observa- 
tory was determined by the Rev. Richard Sheepshanks in 1842 by the transport of 
chronometers from Greenwich, and in 1854 apparatus for the continuous photographic 
recording of atmospheric pressure and temperature was installed by Mr. (afterwards 
Sir) William Crookes. 

From the inception of the Observatory its chief concern had been with fundamental 
meridian work. Though Hornsby had at his disposal an equatoreal sector, and 



146 The Radcliffe Observatory 

10-ft and 3-|-ft Dollond achromatic telescopes which could be wheeled out of the large 
room in the tower on to the balcony, he only used them on occasions for such events 
as eclipses, occultations and the phenomena of Jupiter's satellites ; he obtained 
approximate positions of Uranus with the sector a few days after its discovery by 
William Herschel. And although later on a 10-ft reflector by Herschel, the 
heliometer and a 10-in. equatorial were added to the equipment, the reputation of 
the Observatory rested on its meridian work until that ceased early in the present 
century. 

The meteorological observations, which started with the reading of pressure and 
temperature for the calculation of refraction, developed gradually until a full set of 
observations of all the elements, including the continuous recording of pressure, dry 
and wet bulb temperatures and rainfall, was made. The instruments were naturally 
changed from time to time, and so indeed were their exposures, but it was found 
possible to construct reasonably homogeneous series of values for the elements over 
long periods (for rainfall and mean temperature from 1815 onwards), and these 
results were published in 1932 as an appendix to Volume LV of Radcliffe Observations. 

When Manuel Johnson died in 1859 his place was taken by the Rev. Robert 
Main, who had been Chief Assistant at Greenwich for twenty-five years. He himself 
observed mainly with the heliometer, measuring over 800 double stars and the 
diameters of the major planets. His assistants, at first two and later three, observed 
with the Trottghton and Simms 5-in. transit circle, which had belonged to Richard 
Carrington and was purchased in 1861. This instrument was in regular use until 
1903. Main published the First and Second Radcliffe Catalogues, containing respec- 
tively the results of the meridian observations made between 1840 and 1853 and 
from 1854 to 1861. The Third Catalogue, compiled in Germany from the observa- 
tions made between 1862 and 1876, was not published until 1910. 

Main was succeeded in 1879 by Edward James Stone, who came to Oxford from 
the Cape Observatory, where he had served for nine years as Her Majesty's Astronomer. 
He did not observe himself except with the heliometer, which he employed for deter- 
mining the solar parallax by measuring the positions of minor planets ; but his 
activities were many-sided. He produced the Radcliffe Catalogue for 1890, as a 
continuation up to the equator of the Cape Catalogue for 1880, which had been his 
main concern while at the Cape ; he organized the British expeditions to observe the 
transit of Venus in 1882, training the observers beforehand in Oxford; and he 
successfully observed the total solar eclipse on 8th August, 1896, in Novaya Zemlya. 
In 1886 the Observatory received as a gift from Mr. Gitrney Barclay a very fine 
10-in. equatorial by Cooke, which later ousted the heliometer from its dome and was 
used on occasions as long as the Observatory remained in Oxford. 

On Stone's death in 1897 Arthur Alcock Rambaut, Royal Astronomer of 
Ireland, became Radcliffe Observer. His first work was to introduce some improve- 
ments into the transit circle, and the observations made with this instrument during 
the last ten years that it was in regular use were published in 1906 as the Radcliffe 
Catalogue for 1900. In the meantime the new Grubb twin telescope with 24-in. 
photographic and 18-in. visual objectives had been erected in a dome furnished with 
a hydraulically-operated rising floor. It was at first employed in testing an idea 
suggested by J. C. Kapteyn for determining simultaneously the parallaxes of a 
large number of faint stars. This method was not a success owing to the parallaxes 
being too small in relation to the magnitude equation affecting the results. So in 



H. Knox-Shaw 147 

1909 the telescope was turned to photographing the northern and equatorial Selected 
Areas for the proper motions of stars down to the fourteenth magnitude. There were 
115 such Areas in the programme and the plates were taken in duplicate, one being 
stored for re -exposure after an interval of at least ten years and the other being 
developed at the time for eventual comparison with another plate taken at the second 
epoch. All the first epoch plates were secured in the next nine years, but although the 
efforts of the whole staff were concentrated on this work a further fifteen years were 
to pass before those at the second epoch were completed. 

Rambaut died in 1923, and in the following year Harold Knox-Shaw, Director 
of Helwan Observatory and of the Egyptian Meteorological Service, was appointed 
in his place. The new Observer's first duty was clearly to complete at as early a date 
as possible the observing programme on the Selected Areas and to arrange for the 
heavy computation involved in the derivation of proper motions for 30,000 stars. 
By the introduction of calculating machines and the appointment of two extra 
computers the work was completed in a further ten years, and in 1934 the Radcliffe 
Catalogue of Proper Motions, representing the unremitting labour of the Observatory 
for a quarter of a century, was published. 

The two previous Observers had each wished to reduce the meridian observations 
made by Hornsby, but had been unable to obtain sufficient money for that purpose. 
However, an application now made to the Royal Society was most sympathetically 
received, and generous grants over a period of six years provided the salaries of two 
computers. The work was done in collaboration with Dr. J. Jackson, then Chief 
Assistant at Greenwich, who made himself responsible for the tabular positions of the 
Sun and stars. The results appeared in 1932 as Hornsby's Meridian Observations, 
1774-1798. 

But as is soon to be recorded, the Observatory was ere long to bid farewell to 
Oxford and at the same time to one of its most faithful servants. William Henry 
Robinson, after three years at Greenwich, was in 1879 appointed by Stone to be his 
Second Assistant. He became First Assistant in 1920 and retired in 1935 after serving 
the Observatory with the utmost loyalty and devotion for fifty-six years. 

For some time the Radcliffe Infirmary being in urgent need of room for expansion 
had looked with envious eyes at the Observatory's nine acres next door, and had 
indeed sounded the Trustees on whether they would sell part of the land. It was 
decided, however, that the Observatory could not carry on efficiently if it were 
surrounded by buildings any more closely than it was. So with the backing of its 
new President, Sir William Morris (now Lord Nuffield) the infirmary offered to 
buy the whole site. It was clear that if the Observatory was to move it should be to 
a place with a better climate and preferably to the southern hemisphere. So as the 
British Association was meeting in South Africa in 1929, Sir Frank Dyson and the 
Observer were commissioned to look for a possible site there. Their enquiries met 
with a ready response, and one of the sites offered to the Trust, on the hills outside 
Pretoria, was provisionally chosen ; W. H. Steavenson went out for six months and 
made exhaustive tests of the atmospheric conditions there. These proved so good 
that the generous offer of the Pretoria Municipality to present 57 acres of land to the 
Trust, and to lay on water and electric power, was accepted. The Trustees thereupon 
agreed to sell the whole site to Sir William Morris for the use of the Infirmary and 
the University Medical School, and were granted a lease of the buildings and the 



148 The Radcliffe Observatory 

immediately surrounding land to enable the proper-motion programme to be 
completed. 

There were, however, legal difficulties ahead of the project. The Court of Chancery 
had to sanction not only the sale of the land but also the expenditure of any of the 
capital of the Trust outside its own jurisdiction, and for such an action there was no 
precedent. The scheme providing for the removal to South Africa eventually came 
before the Court in 1934. The Attorney-General was there to see that the interests 
of the general public were not outraged, and the University of Oxford was given 
leave to intervene and plead that the Trust's funds could be put to better use nearer 
home. The Trustees' scheme, which included an offer of close co-operation with the 
Oxford University Observatory, was backed by affidavits from a regular galaxy of 
astronomers : Sir Frank Dyson, Sir Arthur Eddington, Dr. J. S. Plaskett, and 
Professors Willem de Sitter, Frank Schlesinger, and Harlow Shapley, the 
last three testifying to the great value to the home University of a southern station in 
South Africa. The University of Oxford filed affidavits from Professor F. A. Linde- 
mann (now Lord Cherwell) and Professor Albert Einstein. After a sitting of three 
days Mr. Justice Bennett sanctioned the scheme in principle, and it was approved 
in detail a year later. The scheme included provision by the Trustees of funds for a 
Travelling Fellow in Astronomy, who would spend half his time in Oxford and half 
in Pretoria, the selection of the Fellow to lie with the University. 

It was then possible to order from Grubb, Parsons & Co. the 74-in. reflector the 
plans for which had been under consideration for several years. The Oxford site was 
vacated in June, 1935, and the building of the new Observatory was commenced 
early in 1937. The mechanical parts of the telescope were erected in the following 
year, but the large mirror was delayed by two failures in casting the disk and by the 
war, and was not installed until 1948. The Cassegrain spectrograph ordered in 1937 
was not completed until 1951. Though so long delayed the telescope could still claim 
to be the most powerful in the southern hemisphere. 

On the removal to Pretoria new assistants were appointed, who for the first time 
in the history of the Observatory were University graduates; R. 0. Redman, and 
E.G. Williams whose early death was a grievous loss. During the ten years that the 
telescope was without its mirror, observations could be made only through its finders 
and two wide angle cameras lent by the Cape Observatory, which were attached to the 
tube. Williams used the former for determining the magnitudes and colours of early- 
type stars by the Fabry method, and Redman the latter for photometric work on the 
Harvard E regions. He also photographed the change in the solar spectrum on 
transition from chromosphere to photosphere at the total eclipse in 1940. 

With the completion of the telescope the new Observatory was at last able to 
embark on the programme of work which had been planned in outline so many years 
ago. In the forefront of this programme was the measurement of the radial velocities 
of the O and early B type stars in that third of the galaxy too far south to be reached 
from the northern hemisphere. Photography of the far southern nebulae also had its 
place in the original plan. Before this work was well under way Knox-Shaw had 
retired, the first Observer to do so, and had handed over to Andrew David Thackeray. 
He was able to bring into action almost at once a scheme that had been brewing for 
some little time, whereby in return for an annual grant from the British Admiralty 
a third of the observing time with the 74-in. reflector is allotted to staff from the Cape 



C. W. Allen 149 



Observatory. Thus full use can be made of the good climate of Pretoria in a way that 
would not have been possible by the staff of the Radcliffe Observatory alone. There 
is a tremendous field of opportunity open to the new Observatory, and there is good 
reason for thinking that it will be both energetically and successfully explored. 



A System of Quantitative Astronomical Observations 

C. W. Allen 

University of London Observatory 



1. The Problem 

Astronomy is an experimental science, but, since the objects of astronomical inquiry 
cannot be handled or influenced in any way, the experiments of astronomy become 
a matter of observation only. All the practical evidence relating to the heavenly 
objects comes to us in the form of electromagnetic radiations, visible or invisible, 
which may be detected by various methods. An observing astronomer is therefore 
occupied entirely with devices for detecting and measuring these radiations. 

Since it is possible to describe an observer's activities so simply one feels that 
there should be corresponding simplicity about his measurements and results. We 
may introduce some formal simplicity by regarding practical astronomy as a series 
of observations on a number of celestial objects which bear highly significant points 
of similarity and difference. 

The observer's first attention is given to a primary classification of celestial bodies 
by means of their qualitative characteristics. Individual objects are then found to 
have attributes that can be measured quantitatively and the results may be given as 
a set of numbers — one for each attribute. Other attributes are more complex and in 
the first instance can best be represented by a pictorial illustration. However, it 
may be found that much of this illustration can be resolved into a collection of 
entities each to be measured by a further set of numbers. Without doubt further 
analysis could reduce every observable detail to numerical expression but at some 
stage this would become inefficient both to the observer and the user. Thus astro- 
nomical observations become reduced to (a) a quantitative part expressed by sets of 
numbers, and (6) a qualitative part expressed in illustrations or verbal descriptions. 
In many branches of astronomy both parts are needed, the quantitative for com- 
parative analysis, and the pictorial to decide what types of objects exist and to 
provide the ideas. 

We are concerned at present with quantitative measurements which are urgently 
required for all celestial objects in order that an accurate intercomparison might be 
possible. The objects occur in great variety of forms and do not all lend themselves 
very obviously to uniform measurement. There has been no general agreement as to 
how such measurements should be made and each observer has tended to use his 
own system, leading to results that are not directly comparable with others. We 
must, therefore, face the serious question of whether it is in fact possible to make 



150 A system of quantitative astronomical observations 

uniform quantitative measurements of all astronomical objects, and, if the measure- 
ments cannot be made uniformly, what is the best approach to uniformity that 
we can offer. 

We may make some progress with this question by considering the requirements 
of an ideal set of uniform astronomical measurements. With the list of requirements 
before us it should be possible to determine the degree of compromise that must 
be adopted. 

2. Requirements 

(a) The measurements should be within the power of existing observing techniques 
at several observatories and for several objects, but should be capable of improvement 
to any accuracy required by future techniques. 

(6) Any measurement should be available for the whole range of astronomical 
objects without requiring their qualitative classification. It should be suitable for 
describing, measuring, and labelling. 

(c) The measurements should be independent of one another. They should be 
chosen in such a way that each has a useful meaning in its own right and without 
reference to other observations. If a measurement is slightly dependent on another 
factor that factor should be chosen centrally in the range that might exist in practice. 

(d) The set of measurements should be complete in the sense that all available 
general quantitative information should be obtainable from them. 

(e) Each measurement should express a physical quantity and should be given in 
units appropriate to that quantity. They should be readily convertible to analogous 
measurements or quantities. 

(/) Defining rules should be in terms of fundamental physical units and should 
contain no arbitrary factors. If arbitrary numbers are required they should be 
carefully chosen from 2, e, 10, 100, 1000, etc. 

(g) Measurements should possess stability in the sense that if the associated 
quantity is stable the measurement is a constant. They should be on an absolute 
scale but suitable also for differential measurements. 

(h) Measurements should be suitable both for those engaged in specialized researches 
and those engaged in accumulating standard observations. 

If all observations could be made on a scheme that obeyed these requirements a 
very considerable economy might be effected both in observing and expressing 
results. Plentiful observations could be expected each one adding directly to the 
pool available and ready for use in either a theory or statistical analysis. New values 
could be readily compared with old and a mean or "best" value rapidly assessed. 

However, the possibility of developing such a scheme is dependent firstly on 
whether one can set down a list of measurements that can represent the main 
quantitative observations of astronomy. This is not as difficult as it might appear 
and the list below of twenty-five measurements, together with their correlations, 
will be found to be fairly complete. In the list each independent measurement is 
labelled with a number, and the correlation parameters are collectively labelled C. 

3. Measurements 

[1] Number of astronomical objects of particular categories in whole, or a 
defined portion, or the sky. 
[2, 3] Position in the sky. 



C. W. Allen 151 

[4, 5] Total apparent magnitude, and colour. 

[6, 7] Polarization. 

[8, 9] Size and ellipticity. 

[10, 11] Spectrum classification and luminosity class. 

[12] Time. 

[13] Period (of periodic phenomena). 

[14] Duration of phenomena. 

C Variations of [1-11] with time may be expressed in many correlation para- 
meters other than [13, 14]. 

[15] Multiplicity (= number of entities making up whole object). 
[16, 17] Position within object (or departure from mean position) usually relative 
to [2, 3] (includes orientation). 
[18] Surface brightness. 
[19] Number density of entities. 

C Variations of [5, 6, 7, 10, 11, 13, 14, 18, and 19] with [16, 17]. 

[20, 21] Wavelength of a spectrum line, and its shift from normal position. 

[22] Intensity or equivalent breadth of a line. 

[23] Line breadth. 

[24] Line multiplicity. 

[25] Spectral intensity of a continuum. 

C Variations of [20-25] with other measurements. 

We must consider how well this list of measurements suits the requirements. 
The measurements are, of course, not entirely complete, but they cover the main 
factors relating to objects in the sky and their spectra, and include radio astronomy. 
Some measurements cannot be applied in all cases — for example, some objects have 
no clearly defined period for the measurement [13]. However, there are not many 
inconsistencies in the scheme of applying [1-19] to all celestial objects, and [20-25] 
to all spectra. It is mainly technical limitations that prevent us from making all 
measurements on all objects. The items in the list are not entirely independent, for 
example, [5] and [10] or [4, 5] and [25] could give the same information, and [13] 
could be regarded as a correlation of [12] with other factors, but these are rather 
unimportant inconsistencies caused by an attempt to make the list simulate the main 
observations as closely as possible. We might call these the primary measurements 
of astronomy and restrict any further discussion to the domain covered by them. 

The important question to be decided is whether each measurement can be 
adequately expressed by one primary value, and how this can be done by uniform 
observations without loss of vital informations. We take the point of view that, 
whatever other observations might be made for the needs of a special inquiry, the 
listed primary observations should be made in a standardized form on every object or 
spectrum line that is of interest. 

The problems to be faced in standardizing each of these measurements are quite 
varied and must be considered individually. This is done below, and in each case 
an opinion is given as to the most appropriate definition for the primary measurement 
in view of the many facts and requirements. 



152 A system of quantitative astronomical observations 

[1] It is not possible to detect and thence measure the number of objects of any 
particular category in the whole universe nor even in the galaxy. The most useful 
measurement is number of objects down to a selected limiting magnitude. However 
one cannot select one standard limiting magnitude to suit all categories, and for the 
observations to be useful it is desirable to make the two dimensional measurement 
of total number against limiting magnitude. For the most general representation 
the counts should refer to the whole sky, while for regional studies it is more appro- 
priate to make counts per square degree. 

While most celestial statistics will be measured in this form there is scope for a 
single primary measurement to express the populations of different categories as seen 
from the earth. Any such measurements must necessarily be obtained from the 
brighter objects of the particular category since the faint ones often cannot be 
classified. Probably the most direct and stable measurement would be the magnitude 
of the tenth brightest object in the particular category. Such an observation could be 
referred to the population of all objects to the same magnitude (a stable relation 
dominated by stars) to give the proportion of that object relative to all others. If 
the mean absolute magnitude of the object were also known it could be interpreted 
to estimate population in space. 

[2, 3] The measurement of absolute position by right ascension and declination 
offend the stability requirement (g) unless a standard epoch is selected. It would 
appear that 1900-0 shows promise of being adopted as a permanent standard epoch 
for absolute reference of celestial positions. 

[4] The bolometric magnitude would be the ideal standardized measurement of 
apparent magnitude if this were capable of being measured directly. However, 
the practical choice falls between the photographic and photo visual system, of 
which the photovisual appears to have the slight advantage as a primary standard 
because the effective wavelength is more stable and more central among the values 
that might be selected. In the case of extended or composite objects the magnitude 
must refer to the total light in excess of the radiation from the background and 
field stars. 

[5] With the photovisual system chosen as the magnitude standard the colour 
index P — V (where P is photographic and V the photovisual magnitude) becomes 
the natural choice for expressing colours. The weakness of the system is that the 
base-line of colour difference changes with the colour itself, however there is still a 
one-to-one relation between colour index and any other general measurement of 
colour, so that there is every opportunity of standardizing these relations. 

[6, 7] The degree of polarization is unambiguously measured from the ratio 
I x to I y , where / is intensity and the subscripts x and y represent directions at right 
angles for linear polarization and rotations in different sense for circular polarization. 
The ratio might be expressed for preference as (I x — I y )/(I X + /„). The preferred 
conventions for expressing the direction of polarization are (a) for linear polarization, 
the direction in which the electric vector is maximum, (6) for circular polarization, 
the optical convention whereby for R polarization the rotation of a vector in a fixed 
plane when looking towards oncoming light (so that it enters the eye) should be 
clockwise. At present radio astronomers adopt the convention opposite to (b). 

[8, 9] For the primary measurement of size there appears to be every reason to 
use apparent radius (in seconds of arc) for stars, planets, and satellites, but to seek 
another definition for other objects with no sharply defined edge. There are great 



C. W. Allen 153 

difficulties in deciding on a measurement for extended astronomical objects such as 
nebulae and clusters. Not only is there no denned edge but the fall of intensity often 
trails off very slowly and irregularly with the result that size measurements may differ 
by a factor of more than ten. The definition of the edge should be related in some 
way to the brightness of the rest of the object. The central intensity cannot be used 
as standard since it may be an unresolved stellar nucleus, and we are left with the 
most appropriate definition, the diameter of that circle (or ellipse) which will contain half 
the total light. The selection of half for the adopted fraction should give diameters 
comparable with the usual conception of size. It will be noticed that although 
radius is used for stars and planets, diameter is recommended for other objects which 
have no assurance of radial symmetry. 

When objects have measurable ellipticity (i.e. a long and a short axis) it is usual 
in astronomy to quote the major diameter a as a first measurement, and then 
ellipticity (a — b)ja from which the minor diameter b may be obtained. The mean 
diameter for most purposes would be Vab. The procedure for measuring any object 
(not in the star or planet category) would be to decide first on the ellipticity and 
orientation of the major axis. Ellipses of this shape and orientation are then fitted 
to the object and the size varied until it contains half the total radiation. The 
ellipticity may need adjustment to simulate the intensity contours when the final 
size is known. The primary size of the object would be defined by the major diameter 
of this ellipse and its ellipticity. As in [4] only that radiation that is in excess of the 
background radiation should be included. 

[10, 11] The two-dimensional system of spectrum and luminosity classification is 
maintained by standard stars. However a certain amount of classification error 
should be allowed for these standards so that they do not prevent the establishment 
of smooth relations between spectral class and other factors such as colour. To 
establish such relations one must know whether the domain of classification extends 
from 0-0 to 9-9 for each class (O to M ) or whether there are some sections definitely 
missing. Existing systems (mainly the Yerkes classification) are best illustrated 
graphically when the domain K8-0 to MO-0 is completely omitted. It would be 
preferable, however, if the K classification could be adjusted to cover the whole range 
as for the other classes. 

[12] The U.T. will remain the primary measurement of time, although the actual 
observations might be made against a sidereal time clock, or the result may be 
converted to ephemeris time, heliocentric time, etc. for special purposes. 

[13] The representation of periods in seconds, days or years presents no formal 
difficulty. 

[14] The measurement of the duration of a phenomenon that commences or ends 
suddenly is quite straightforward, but difficulties arise when one attempts to devise 
a scheme that will give full duration for such events and still give an appropriate 
value for a phenomenon that commences or ends gradually or sporadically. A rule 
that will suit all cases can hardly be expected but some satisfaction may be obtained 
from measurement of the interval between the two instants (a) when 0-1 of the total 
flux of the phenomenon has passed, and (6) when 0-9 of the total flux has passed. 
However, these ratios are too arbitrary for a primary definition and it would appear 
that for this measurement the definition itself will have to be varied to suit the 
circumstances. The measurement of the interval for which the phenomenon exceeds 
an arbitrary threshold level will be used in many cases. 



154 A system of quantitative astronomical observations 

[15] The measurement of the multiplicity of an object is formally equivalent to 
[1], the counting of objects. Again there is no way to select a standard limiting 
magnitude (or limiting resolution) and usually the multiplicity is expressed by giving 
the number of entities as a function of some limiting factor that controls their 
detectability (such as the magnitude). It is only when the multiplicity converges to 
a finite value as the other factor is extended indefinitely that a stable primary value 
of multiplicity can be given. 

[16, 17] These measurements can represent either (a) the displacement of the object 
from some mean or standard position, or (6) the measurement of components of the 
object from some fiducial position. The primary co-ordinates would be the radial 
distance (usually in seconds of arc from the central fiducial position represented by 
[2, 3]), and the orientation (anticlockwise from iV) at epoch 1900-0. 

[18] The practice of expressing surface brightness as an apparent magnitude of a 
defined area has the advantage that great differences of brightness can be expressed 
without change of units. This being so it is desirable that the unit of area be 
selected once for all, and the square second of arc would appear to be the suitable 
choice for primary measurement. However, when components of radiation have 
to be added the logarithmic magnitude scale is inconvenient and one might 
adopt as a second choice the number of m v = or m v = 10 stars per square 
degree. 

[19] The measurement of number density of entities or components of an object 
is similar to the measurement of its multiplicity [15]. 

[20] The custom of expressing wavelengths shorter than A 2000 in vacuum units 
and those longer than A 2000 normal dry air at 15°C is likely to be continued. In 
the radio spectrum frequencies are more usually adopted for accurate work and the 
ambiguity avoided. 

[21] No ambiguity is involved in expressing wavelength shifts in Angstroms or 
kaysers (cm -1 ). 

[22] The intensity of an emission or absorption line that is associated with a stable 
continuous spectrum may be expressed as an equivalent breadth of that continuum. 
This is convenient both for measurement and analysis. However, if no continuum is 
present, measurements must be given directly in physical units, either absolute or 
relative. For the sake of uniformity it would be preferable to express primary 
absorption line intensities as equivalent breadths, and emission intensities in physical 
units such as erg cm _2 s _1 . 

[23] Line broadening is usually symmetrical and regular and hence its measure- 
ment does not involve the difficulties of [14], which is analogous. The accepted 
primary measurement is the total wavelength distance between the points where the 
intensity is half the maximum. 

[24] The measurement of line multiplicity in astronomical sources presents no 
difficulty provided the spectrographic resolution is finer than the line width. Com- 
ponents that would have been resolved if the line width were much smaller are not 
to be regarded as adding to the multiplicity. 

[25] Absolute measurements of spectral intensity / A may readily be expressed in 
physical units such as erg cm~ 2 s _1 A -1 . However, actual measurements are usually 
relative and it is recommended that these should be expressed in the same physical 
units relative to the value at 5400 A. Such measurements could be readily related 
to visual magnitudes. 



H. Bondi 155 

Reviewing this discussion we find that standard practice provides suitable primary 
measurements in most cases. A new measurement to represent the population of 
objects seen from the earth is suggested in [1], and a suggestion for standardizing 
the size of nebulae, etc., is revived in [8, 9]. On the other hand, we have found no 
suitable primary measurement of the duration of irregular events. 

If astronomers in the course of their many and varied observations could add as 
many as possible of the above primary measurements to their programme the results 
would always be welcome. In particular, if the work has other intentions the primary 
measurements should be added as a by-product. There are, of course, many other 
primary factors (such as distance) which are also needed from observers and which 
fall outside the scheme of observations considered here. But the same arguments 
a PPly — if a clear-cut, unambiguous, and well-accepted method of expressing the 
factor is available there will be much better prospects of obtaining usable results. 



Fact and Inference in Theory and in Observation 

H. Bondi 

Trinity College, Cambridge* 
Summary 

A critical examination is made of the relative reliabilities of theoretical and observational work in 
astronomy. It is found that, contrary to widely held opinions, observational papers are no less liable 
than theoretical papers to reach erroneous conclusions. 



The aim of the present paper is an examination of the relative reliabilities of so-called 
"theory" and so-called "observation" in astronomy. They may simply be regarded 
as different tools used by research workers. Such examinations of relative reliabilities 
are common enough in all fields of science. Measurements made by different instru- 
ments are generally found not to agree amongst themselves, and the subject of the 
combination of observations attempts to deal rationally with such difficulties. 
Statistical weights are used to describe the relative reliabilities of similar instruments. 
In the event of a contradiction between two instruments it is helpful to have some 
knowledge of the degree of certainty to be assigned to them. In ordinary laboratory 
practice this assessment is generally based chiefly on past experience with the instru- 
ments concerned, combined with comparisons of their constructions and estimates of 
how "direct" the measurements are. 

This purely empirical approach has been found far more satisfactory in laboratory 
work than reliance on emotional prejudices, however real they may seem without 
critical examination. Empiricism is vital to all scientific progress, and there seems to 
be good reason to apply this empirical approach, which has been so successful in the 
laboratory, to a rather wider field. 

It is fashionable nowadays to divide astronomical research into two categories, 
"theoretical" and "observational". These terms are ill-defined and lack sure philo- 
sophical significance, but it is nevertheless true that most astronomers would have 

* Now at King's College, University of London. 



156 Fact and inference in theory and in observation 

little doubt or disagreement on how to classify most published papers. All references 
in this essay to "theory" and to "observation" are intended only in this colloquial 
sense and should not be understood as more than a rough and ready classification. 

There is undoubtedly a widespread opinion that astronomical theory is, by and 
large, airy speculation, that it changes from day to day in its most fundamental 
tenets, that it is based on ill-considered hypotheses and that its reliability should 
accordingly be assessed to be low. By contrast the results of observational work are, 
according to this opinion, solid incontrovertible facts, permanent and precise 
achievements, that will never change and whose reliability is accordingly high. 

In my view these opinions are unfounded and false, and their prevalence does great 
harm to the progress of astronomy. If they were not injurious, there would be little 
point in attempting to dispel these views, but, in the present state of astronomy, 
conflicts between observation and theoretical results continually arise. In the past, 
credence in such a conflict has generally been given to the observational result, and 
theoretical advances of considerably significance have been held up by undue reliance 
on the observations. The chief purpose of this paper is to show that in such a conflict 
it is far wiser to keep an open mind or even to lean to the side of the theory. Such an 
attitude, which can be firmly based on a critical examination of past experience, is 
likely to be far more helpful to the progress of astronomy than reliance on prejudices 
based on the emotional significance of such outworn phrases as "observational fact" 
and "theoretical speculation". 

Before an examination of the relative reliabilities of these so-called theoretical 
and observational results can be attempted, it is desirable to show that theoretical 
work falls into two quite distinct classes. One is essentially an attempt to classify 
observational material ; the ad hoc assumption is made that the type of classification 
is based on intrinsic qualities without making these particularly clear or (and this is 
the chief characteristic of this type of structure) linking up these qualities with 
known phenomena. An example of this type of structure is the old Russell "theory" 
of the luminosity -type diagram. As will be remembered the principal suggestion of 
this theory was that the different appearance of giant and dwarf stars was due to the 
fact that they were made of different materials, "giant stuff" and "dwarf stuff" as 
the terms were. Any further explanation beyond this, any link with the known 
properties of material was left for the future. As is well known the suggestion was 
later disproved, but it is a typical example of such suggestions. 

An example of more recent date concerns the ingenious observation by McKellar 
(1949) of the abundances of carbon isotopes on certain very red giant stars. He 
suggested that the stars in which the isotope ratio agreed with that produced in the 
Bethe cycle were well mixed stars, and those where the ratio was different, were not 
mixed. The sole purpose of any such suggestion can only be merely to fit the immedi- 
ate observational classification. Indeed it is little more than a re -wording. To anyone 
who has the slightest knowledge of stellar structure, McKellar's suggestion can 
only sound utterly unconvincing, since it raises many more questions than it solves. 

An entirely different type of logical structure to which frequently the same name 
of "theory" is given, is in fact a development of terrestrial physics. An attempt is 
made to apply the laws of physics as obtained in the laboratory to the conditions of 
the astronomical problem in question. Naturally it is sometimes necessary in the 
present state of knowledge to make a number of additional assumptions and the value 



H. Bondi 157 

of the attempt will depend greatly on the clarity and explicit nature of these assump- 
tions. In my view misunderstandings have frequently arisen through confusion 
between the two types of approach both of which have been referred to as "theory". 
Different names are required, and I suggest "observational inference" for the first 
type of structure, and "physical theory" for the second type. An example of this 
second type of structure is the theory of stellar constitution. 

In attempting to assess the relative reliabilities of theoretical and observational 
work in the past it is necessary to establish a criterion of failure. This is not quite as 
simple as one would expect it to be. 

It is an essential feature of science that everything in it may stand in need of 
revision as a result of new and unexpected evidence — that it contains no "ultimate 
truths". Such truths are certainly outside the scope of science, though they may 
appertain to philosophy and to religion. Whenever we examine any approach to 
science we will not be surprised to find that changes occasionally occur, and this is 
true both of theory and of observation. Some such changes arise indeed merely from 
a widening of the field of enquiry. Classical mechanics was found wanting when it 
was applied to atomic dimensions and had to be replaced by quantum mechanics. 
The validity of classical mechanics in the macroscopic field is, however, not affected 
by this change. The changes that are to be examined here, the errors that should be 
taken account of, are of a far grosser and less subtle kind. They are simply statements 
that were later shown to be incorrect in the very field to which they were originally 
intended to apply within the accuracy originally claimed. Such errors have almost 
always been a hindrance rather than a help to the progress of science and we shall 
understand by the word error this its usual conventional meaning (as, for example, in 
the expression "probable error") without in any way entering upon the meta- 
scientific problem of a definition of the term. 

Three questions seem then to arise : 

(i) Is observational knowledge by its very nature more or less certain than 
theoretical knowledge ? 

(ii) As an empirical test, have theories in the past been shown to be in error more 
or less frequently than observational results ? 

(iii) Is an error in observation likely to persist for longer or shorter than an error 
in a theory ? 

Where the first of these is concerned the extreme complexity of present-day 
observational methods hardly need be stressed. To derive any significant astronomi- 
cal result from the blackening of a photographic plate or the simple reading of a 
meter a tremendous amount of intervening work has to be done. Corrections may 
have to be applied, calculations and reductions may have to be carried out, and 
above all interpretations requiring a great deal of theoretical background may have 
to be made. Consider, for example, such an apparently straightforward matter as 
the determination of the masses of an eclipsing binary. Gravitational theory, tidal 
theory, theoretical reflection factors, and other theoretical notions have all to be 
brought in and used, frequently to the limits of their power. Or consider work on 
spectroscopic abundances where not only the most accurate photometry, but also 
particularly complex aspects of quantum theory are involved. Nobody can fail to 



158 Fact and inference in theory and in observation 

admire such work, but by no stretch of imagination can it be termed, purely factual 
or purely observational. And yet this is what some people seek to do ! 

In a recent review of Mr. Hoyle's broadcast talks by Dr. Williamson (1951), a 
review possibly written more in anger than in earnest, the reviewer states that he has 
attempted to establish the percentage of astronomical "facts" as compared with 
"theory" in Mr. Hoyle's talks. But what is an astronomical fact ? At most it is a 
smudge on a photographic plate ! Does he expect Mr. Hoyle to give a broadcast 
talk on smudges ? 

The purely factual part of the vast majority of observational papers is small. It 
is also important to realise that these basic facts are frequently obtained at the very 
limit of the power of the instruments used, and hence are of considerable uncertainty. 
To refer to observational results as "facts" is an insult to the labours of the observer, 
a mistaken attempt to discredit theorists, a disservice to astronomy in general and 
exhibits a complete lack of critical sense. Indeed I would go so far as to say that this 
sort of irresponsible misuse of terminology is the curse of modern astronomy. 

Present-day observational astronomy may fairly be called the science of extracting 
the maximum information from the fundamentally meagre data that can be obtained 
about outer space, an endeavour to stretch both observation and interpretation to 
the very limit. 

A similar examination of physical theories in astronomy reveals that their primary 
basis is very sound indeed, since it rests on established terrestrial physics. But in 
order to apply this knowledge to astronomy inferences of considerable range have 
to be made, sometimes with the aid of additional assumptions. In the theory of 
stellar structure, for example, results obtained in thermodynamic systems of tempera- 
tures of at most 4000° or in non-thermodynamic systems (particle accelerators) 
employing extremely tenuous matter have to be applied to dense matter at millions 
of degrees. It is only because of the comprehensive nature of laboratory physics that 
such extrapolations are possible. Other examples of a similar nature could be given, 
but it would probably be fair to sum up the situation by saying that in observational 
work long chains of inferences are based on frequently somewhat uncertain data, whereas 
in physical theories of astronomy, though long chains of inferences are also used, they are 
generally based on much more reliable experimental data. There is, therefore, no reason 
to expect any marked difference in the degrees of reliability of so-called theory and of 
observation, unless indeed it is that theoretical results are of greater reliability. 

We can now turn our attention to the second question, that of empirical test. The 
question is, have theories been disproved more or less frequently than observational 
results? Clearly it is difficult to give an exhaustive list on either side. On the 
theoretical side I might mention Jeans' proof (1925, 1927) that stars would be un- 
stable if the subatomic generation of energy depended very sensitively on tempera- 
ture, a proof that confused the development of the theory of the constitution of the 
stars until Cowling (1934, 1935) showed it to be fallacious. Again Jeans' "long 
time scale" for the age of the galaxy (1929) was widely accepted until it was disproved 
by many arguments (see Bok, 1946). Finally, one might mention Eddington's 
estimate of the hydrogen content of the stars as about 35 per cent (1930). This last 
error must be shared between theoretical and observational astronomy, since spectro- 
scopists were of the same opinion. The recognition that the hydrogen content was 
far higher came more recently (Dunham, 1939; Hoyle, 1947). 



H. Bondi 159 

Although in my own work I am more in contact with theoretical than with 
observational research, yet I find that I have more often met observational errors. 
One might recall van Maanen's observations of the proper motions in extragalactic 
nebulae (1916, 1921, 1922, 1923, 1925, 1927), observations that have been proved to 
be incorrect not by factors of 2 or 3, but factors of 100 or 1000 (Hubble, 1935). 
Then one might mention Adams' "discovery" of the Einstein shift of the spectral 
lines of Sirius B (1925). This shift is now not only considered to be practically impos- 
sible to measure, but is known through modern quantum theory to be very seriously 
obscured by almost incalculable pressure shifts (Lindholm, 1941 ; Adam, 1948). 
But the shift "observed" in 1925 was supposed to agree as well as could be expected 
with relativity theory without account being taken of these pressure shifts. 

Yet another example in this field is formulated by the Trumpler stars (Trumpler, 
1935; An unjustified interpretation of an uncertain and difficult observation was 
widely accepted as an established fact, as a proof that extremely massive stars 
existed with known luminosities and radii that were apparently in contradiction 
with the theory of stellar structure. Confusion was caused by reliance on this result, 
but now it is regarded as having been based on erroneous interpretations (Struve, 
1950). 

In an earlier period, astronomers in this country, after failing to discover Neptune, 
endowed this planet with a ring and satellites (Lassell, 1847). More recently, 
Hubble and Humason (1931) inferred from their data that the constant of the red- 
shift of the nebulae was 4-967 ^ 0-012 and soon afterwards, from almost the same 
data, that it was 4-707 ± 0-016 (1931, 1934; Hubble, 1936). Similarly the last 
determination of the solar parallax (Spencer Jones, 1941) is well outside the three- 
fold stated probable errors of earlier determinations which vary amongst themselves 
far more than their individual stated errors. 

In a field with which I have been in close contact, Struve 's recent work on Capella 
(Struve, 1951) has come as something of a shock. We were told in the most meticu- 
lous survey (Kuiper, 1938) that the masses of the components of Capella were the 
second best known ones, that, except for two cases, all other mass determinations were 
at least twice as uncertain as Capella, and, except for four other cases, more than 
three times as uncertain. And now it appears that even in the case of Capella there 
has been all along an error of more than 25 per cent. Eddington in particular used 
Capella as his standard when he developed the theory of stellar structure. More 
recently, but for Struve 's timely discovery of the mistake, the subject of the 
structure of Red Giant stars would have been put on the wrong track by reliance on 
this supposedly so precise and permanent fact. 

These examples, though by no means exhaustive, will illustrate sufficiently the 
thesis of this essay. It seems that, by the empirical test, errors in theories are if 
anything less frequent than in observational work. A detailed numerical test is 
difficult, but what I have said is enough to refute the view that theories are airy 
pieces of guesswork disproved every few days whereas observational results are 
"hard", "incontrovertible" facts. 

We come then to the third question : are errors likely to persist for longer in theory 
or in observation ? The answer to this question is clear. Observational equipment 
is so scarce, is devoted to so many tasks, and is so difficult to set up, that repetitions 
and checking are not as common as one would hope, particularly in the case of many 



160 Fact and inference in theory and in observation 

theoretically important measurements. There is also a widespread opinion, an opinion 
that I hope will not persist, that the certainty of observational work is so great that 
no repetitions are required. This uncritical attitude is greatly to be deplored. If 
more effort were devoted to repeating observations, the gain in certainty would be of 
great value. 

The attitude and climate of opinion in which theoretical astronomy is conducted 
are entirely different. Almost every paper is received with sceptical interest and 
many papers immediately stimulate work connected with their proof or disproof. 
While in observational work it is unfortunately considered somewhat impolite for 
one observer to criticize the observations and immediate inferences of another 
observer, similar criticism between theorists is luckily considered perfectly natural. 
There is therefore a considerable likelihood of an error being rectified speedily. As a 
recent example of this I might mention a paper by Richardson and Schwarzschild 
(1948) & Schwarzschild (1948) on Red Giants which was refuted by Gold (1949), his 
paper being submitted less than two months after the publication of the first paper. 
An error in a paper on stellar structure by Chandrasekhar and Shoenberg (1942) 
was shown up speedily by Hoyle and Lyttleton (1946). Of course, sometimes the 
rectification of an error takes much longer. Jeans' statement (1925) about the in- 
stability of stars with temperature sensitive energy production was finally disbelieved 
only after Cowling's work in 1934, twelve years after the claim had appeared. But 
even this interval is not long compared withthe cases quoted in observational 
astronomy where the intervals are usually more like twenty years. The intricate 
nature of observational equipment makes intervals of several years almost un- 
avoidable, but the great lengths of interval occurring are probably due to the 
unfounded prejudice of regarding many observational results as facts not requiring 
confirmation. 

The persistence of observational errors being generally much greater than of 
theoretical errors implies that if the average fraction of errors produced in the two 
branches is roughly equal then the fraction of incorrect current observational work 
is considerably greater than the corresponding fraction for theoretical work. Careful 
statistical analysis would be required to confirm this statement and put it into 
quantitative form, but the arguments given here seem to support this conclusion 
strongly. 

So far attention has been confined to physical theories rather than observational 
inferences, since there is a considerable structural difference between them from the 
point of view of scientific methodology. The fate of observational inferences is not 
encouraging to anyone thinking of relying on this method, many of them having 
turned out to be quite incorrect. Nevertheless they still continue to be made, mainly, 
though not entirely, in the context of observational papers. This seems to be the 
result of a deep human prejudice, that if only one continues to look at an object for 
long enough its nature will become apparent. In science this is, of course, nonsensical. 
One could stare at a piece of wood for years if not generations without discovering 
its atomic nature, or being able to infer its properties in any way from appearances. 
I must refer again to Dr. Williamson's article (1951) which is so usefully revealing 
in presenting current prejudices without any attempt at veiling or rationalizing 
them. He clearly considers it to be a valid argument that people who have never 
done actual observational research are not entitled to discuss astronomy. A more 



H. Bondi 161 

preposterous statement is hard to imagine. It is on the same plane as the statement 
that only plumbers and milkmen have the right to pronounce on questions of 
hydrodynamics . 

As an example of an observational inference one can quote from Strtjve's book 
(1950, p. 116) : "It looks as though the K-type and M-type dwarfs represent some- 
thing in the nature of the final stage. . . . This part of the H-R diagram resembles 
a sink into which many stars drop . . .". 

This is not the place for discussing such an idea in detail but I cannot help sus- 
pecting that the feeling described is at least partly due to a prolonged study of 
Hertzsprung-Russell diagrams drawn in the conventional way with the red 
dwarfs near the bottom of the picture. 

The final point I wish to make concerns stated probable errors. All too often 
subsequent work has shown that they bear little relation to the actual errors made 
but are at most an indication of the internal consistency of the methods used. It 
would be a tremendous help if more observational papers were to contain (as some do 
now) a reasonable assessment of the errors that may have arisen. On the theoretical 
side it would be similarly of great advantage if more papers could contain clear 
explicit statements of the assumptions made, of every appeal to observation, and of 
every subsidiary hypothesis. Then the observational disproof of a theory would 
convey immediately valuable information. It would become clear that one of the 
bases of the theory was wrong, and such discoveries have frequently been very 
valuable, as for example, when the Michelson-Morley experiment showed that the 
velocity addition formula underlying the original theory was wrong. If these 
habits of clearly stating uncertainties and assumptions became general, and if 
prejudices regarding observational results as facts and theories as bubbles were 
overcome, astronomy would greatly benefit. Both so-called theory and so-called 
observation are liable to error, and critical appreciation and impartial scepticism 
are the best foundations for progress. 



RErEBENCES 

Adam, M. G 1948 M.N., 108, 446. 

Adams, W. S 1925 Proc. Nat. Acad. Set., U.S.A., 11, 382. 

Bok, B. J 1946 M.N., 106, 61. 

Chandrasekhae, S. and Shoenbebg, M. . . . 1942 Ap. J., 96, 161. 

Cowling, T. G 1934 M.N., 94, 768. 

1935 M.N., 96, 42. 

Dunham, T. F., Jr 1939 Proc. Amer. Phil. Soc, 81, 277. 

Eddington, A. S 1930 The Internal Constitution of the Stars, 

p. 159 (Cambridge). 

Gold, T 1949 M.N., 109, 115. 

Hoyle, F. and Lyttleton, R. A 1946 M.JV., 106, 525. 

Hubble, E. and Humason, M. L 1931 Ap. J., 74, 43. 

1934 Proc. Nat. Acad. Sci., U.S.A., 20, 264. 

Hubble, E 1935 Ap. J., 81, 334. 

1936 Ap. J., 84, 270. 
Jeans, J. H 1925 M.N., 85, 914. 

1927 M.N., 87, 400, 720. 

1929 Astronomy and Cosmogony, p. 381 (Cam- 
bridge). 

Kuipeb, G. P 1938 Ap. J., 88, 472. 

Lassell, W 1847 M.N., 7, 157, 167, 297, 307. 

Lindholm, E 1941 Ark. Mat., Astron. Fys., 28B, No. 3. 



162 Philosophical aspects of cosmology 

Maanen, A. van 1916 Ap. J., 44, 210. 

1921 Ap. J., 54, 237, 347. 

1922 Ap. J., 56, 200, 208. 

1923 Ap. J., 57, 49, 264. 
1925 Ap. J., 61, 130. 
1927 Ap. J., 64, 89. 

McKellar, A 1949 Publ. Astron. Soc. Pacific, 61, 199. 

Richardson, R. S. and Schwarzschild, M. . . 1948 Ap. J., 108, 373. 

Schwarzschild, M 1948 Ap. J., 107, 1. 

Spencer Jones, H 1941 M.N., 101, 356. 

Strtjve, 1950 Stellar Evolution, pp. 18-20 (Princeton). 

1951 Proc. Nat. Acad. Sci., U.S.A., 37, 327. 

Trxjmpler, R, J 1935 Publ. Astron. Soc. Pacific, 47, 254. 

Williamson, R 1951 J. Roy. Astron. Soc. Canada, 45, 185. 



Philosophical Aspects of Cosmology 

Herbert Dingle 

Department for History and Philosophy of Science, University College, London 



1. The purpose of this paper is to indicate, but not to solve, certain philosophical 
problems peculiar to the study of cosmology. Cosmology and cosmogony — no attempt 
is made here to distinguish them, and they are in fact inseparable — are at the present 
time primarily scientific subjects ; that is to say, we are no longer forced to approach 
them on grounds of pure reason alone but have in our possession a growing body of 
observed facts which must form the data on which our reason begins to operate. 
Nevertheless, they have the peculiarity that in them we treat the whole field of 
investigation as having characteristics of its own, independent of the characteristics 
of any of its parts, and it is those universal characteristics that we seek to discover. 
But, up to the present at least, the part of the universe that we can observe is at most 
only a very small portion of what we have reason to believe exists. We have therefore 
to introduce considerations over and above the ordinary scientific process of inductive 
generalization, and those considerations are philosophical in character and so give to 
cosmology a philosophical aspect which the other departments of science do not show 
in the same degree. 

An example will make the point clearer. When Newton demonstrated that within 
the solar system the movements of bodies everywhere conformed to the law that 
every piece of matter attracted every other piece of matter with a force varying 
directly as the masses of the bodies and inversely as the square of the distance 
between them, this law was generalized to apply to matter everywhere, and so 
became known as a universal law. But it was not a law of the universe ; it was a 
law that, supposing it to be true, was exemplified wholly and completely in every 
part of the universe, but it had nothing to do with the universe as a whole. The 
universe might be large or small, finite or infinite, eternal or temporary, homogeneous 
or heterogeneous — in fact, it might, as a whole, have any conceivable characteristics 
at all, and the Newtonian law of gravitation would be the same in all cases. Similarly, 
the laws of thermodynamics are universal laws but not laws of the universe. Again 
assuming them to be true, they characterize any closed system whatever, small or 
large, irrespective of whatever else the universe might contain ; we therefore learn 



H. Dingle 163 

from them nothing at all about the character of the universe as a whole. On the other 
hand, when you say that space has a positive or a negative or a zero curvature, you 
are stating a law of the universe, but you are saying nothing at all about any particu- 
lar part of the universe. The curvature in a given region may be anything ; it is 
only the universe as a whole that is supposed to have the curvature you postulate. 
Accordingly, the assertion that the universe has this curvature must rest on something 
other than direct generalization from observation or experiment in limited regions ; 
it must involve reasoning over and above the ordinary type of scientific reasoning. 
It is the problems connected with that kind of reasoning that are most properly 
called philosophical problems of cosmology. 

2. By its very definition the universe is necessarily unique. The first question that 
arises, therefore, is whether its uniqueness is, so to speak, essential or accidental. 
That is to say, can we properly imagine different kinds of universes that might have 
been formed, and so present ourselves with the problem of showing why it was in 
fact the actual one that came into existence ; or must we take it as a primary axiom 
that any conceivable alternative to the actual universe must necessarily be impossible ? 
This question must be answered before we can decide whether or not familiar scientific 
methods of attack are legitimate when applied to the universe as a whole. For ex- 
ample, in the statistical mechanics of Gibbs, when we wish to study the behaviour of 
a sample of gas, we first of all suppose the sample to be composed of a very large 
number of molecules, each behaving in an unspecified — or at most an incompletely 
specified — manner, and then consider a very large number of such samples. Owing to 
the incomplete specification we cannot deduce with certainty how this ensemble of 
samples will behave, but we can deduce its probable behaviour. We then draw 
conclusions, from this probable behaviour of the ensemble, about the actual behaviour 
of the sample in which we are interested. 

Now whatever philosophical doubts may be aroused as to the validity of deduc- 
tions made in this way, there can be no doubt concerning the possibility of making 
the investigation. Undoubtedly a very large number of samples of gas of the kind 
postulated do exist, and undoubtedly, if they are composed of molecules, we must 
leave unspecified the instantaneous positions and momenta of those molecules and 
therefore have no right to assume that such positions and momenta in the various 
samples bear any relation to one another. But have we any right to apply the same 
kind of treatment to the universe ? Can we consider an ensemble of universes, in 
each of which the stars and nebulae can form any configuration whatever, and then 
draw conclusions about our actual universe from the probability of behaviour of this 
imaginary ensemble ? It is at least a plausible proposition that we have no right to 
talk of the probability of an event before we are satisfied that the event is possible. 
We have, so far as I know, no assurance that any universe other than one is possible. 

3. Consider now another example — not an imaginary one, but one that has revealed 
itself in the recent history of cosmology without, apparently, arousing the philo- 
sophical questions that should have been asked. The notion of "stability of equili- 
brium" is a very familiar one in mechanics. A system is said to be in equilibrium 
when, in the absence of disturbances from the outside, its state remains unchanged 
for an indefinite time ; the equilibrium is said to be stable when, if the system is 
momentarily disturbed by an indefinitely small external impulse, it automatically 



164 Philosophical aspects of cosmology 

tends to return to its former state ; if, on the other hand, its reaction to such an 
impulse is to remove itself further from its former state, it is said to have been in 
unstable equilibrium. Without the external impulse we cannot tell whether the 
equilibrium is stable or unstable ; it is simply equilibrium. Now suppose we are con- 
sidering the universe, and suppose we have reason to think that it is in equilibrium ; 
how can we tell whether its equilibrium is stable or unstable ? Only by disturbing 
it from the outside. But by hypothesis there is no outside ; the universe, whether 
finite or infinite, comprises all that is. Hence the distinction between stability and 
instability ceases to have any meaning when applied to the universe. Now it is 
well known that, according to the cosmology of the late Sir Arthur Eddington, the 
universe was once in equilibrium ; it was the static Einstein universe familiar in the 
early days of the relativity theory. But Eddington showed that this universe was 
unstable — that is to say, if it was disturbed it would not return to its original state 
but would start and continue to expand or contract — and in this way he accounted 
for the present observed recession of the extra-galactic nebulae. But what, then, 
disturbed it ? If something outside, the system considered was not the universe ; if 
something inside, the system considered was not in equilibrium. Actually Eddington 
chose the latter alternative, but he did not face the problem of explaining how the 
original universe could have been both in equilibrium and not in equilibrium. Yet 
the problem remains, and is clearly not one to be settled by observation; it is a 
problem of the philosophy of science. 

4. A more general aspect of problems of this kind is presented by the following 
question : can we, in a description of the universe, maintain the separation that has 
been implicit in science at least from the seventeenth century onwards, between 
universal laws of nature and the particular material system that actually exists and 
behaves in accordance with those laws ? It is probable that, at any rate up to very 
recent times, few would have questioned the legitimacy of this separation. For 
instance, suppose it had been asked : "If there had been another major planet between 
Mars and Jupiter, would the law of gravitation have been different?" The almost 
unanimous answer, I think, would have been: "No, the law of gravitation is true 
independently of the actual bodies which happen to be in the universe. The intro- 
duction of another planet would affect in some degree the motion of every body in 
the universe, but the new motions would still obey the same law of gravitation". 
The reverse question: "If the law of gravitation had been different, would the 
number of bodies in the solar system have been different ?" would, I think, have been 
regarded as scarcely permissible ; the law of gravitation would have been accorded 
a stronger element of necessity, so to speak, than the actual population of the 
universe. But if the question had been pressed, the answer would probably have 
been that one could not say with confidence, but that probably there would have 
been a difference. The process of formation of the planets, taking place according 
to a different law, would certainly have been different in detail, and that might very 
well have affected the number of planets formed. The general picture was that of 
eternal and unchangeable laws, independent of any material creation that might 
or might not exist to display them ; and superposed on that, a particular arrangement 
of bodies, whose number and disposition must result from the particular process 
that brought them into being. 

Such a picture has proved invaluable for the study of isolated systems — that is, 



H. Dingle 165 

systems bounded in space and time. We choose an arbitrary moment at which we 
observe their state, and then calculate what the universal laws will do with bodies 
in that state; and in innumerable instances our calculations are justified by the 
event. It is therefore tempting to apply the same way of thinking to the problem of 
the universe. But is this justifiable ? Apart from the impossibility, which relativity 
has revealed, of uniquely associating a particular spatial boundary with a particular 
instant of time — an impossibility whose effects become more and more serious the 
larger the system we study — we find that we must choose our "arbitrary" instant of 
time at the beginning of things, since the further out in space we go, the earlier is the 
time at which we must necessarily observe the state of the bodies there. In other 
words, we have no longer, in fact, an arbitrary instant to start from ; the moment of 
creation itself, if that is the proper name for it, is the only possible one. Similarly, 
our spatial "boundary" must be removed to an indefinitely great distance, for we 
cannot screen off a portion of the universe from the rest and generalize from that. 
We are thus forced to start at the origin of the whole of things, and how can we be 
sure that the laws which we deduce from the contemporary behaviour of small 
regions are necessarily the laws that operated everywhere then ? 

5. The question is closely bound up with the view we take of the source of our 
knowledge. The traditional scientific belief has been that both the laws of nature and 
the structure of the material system that is the visible body of nature are knowable 
only through observation, through experience. If that is so, then it becomes very 
difficult to maintain that the laws are essentially independent of the structure. The 
observed behaviour of the universe now is the single source of both elements of our 
knowledge ; if that were different — as we all believe it would be if the configuration 
of bodies were different — the proposition that we would infer the same laws becomes 
very dubious. And, indeed, the modern relativistic approach no longer assumes the 
old independence of laws and structure. Relativistic cosmology gives us in the same 
formula the structure of a system and the motions occurring in it, and we cannot 
have one without the other. The Einstein universe contains so much matter, and 
it is necessarily static ; the de Sitter universe is expanding, but necessarily has no 
matter in it; the Lemaitre universe is expanding, and necessarily has a homo- 
geneous distribution of matter having certain characteristics and no others. This is 
quite consistent with a purely empirical approach in which we make no presupposi- 
tions but describe as a whole the kind of universe we observe. A separation between 
laws and structure would indeed be plausible, if not inevitable, if one had a rational 
and the other an empirical basis — if, that is to say, the laws of nature, but not the 
appearances of the constellations, could be derived by reason without recourse to 
experience. This, in fact, was maintained by Eddington, and on other grounds by 
Milne, though Eddington at least accepted the relativity principle that bound the 
two inseparably together. It is somewhat paradoxical that when such a rational 
origin for the laws of nature would have justified the separation that was universally 
assumed, no scientist admitted it, and now that the separation is discredited, cos- 
mologists appear who hold a view that would demand its acceptance. Here is perhaps 
a philosophical problem of psychology, but cosmology affords scope enough for our 
present troubles. 

6. Space forbids a consideration of the problems concerning time and the origin of. 



166 Modern cosmology and the theologians 

the universe. It must suffice to mention, and to question, one assumption that has been 
tacitly made in some recent theories, namely, that, pravrded^ahynothesis is consistent 
with experience and contains no self-contradictions, it is valid, and therefore that 
anything postulated to have happened before human experience began is exempt 
from the requirement that it must conform to our possibilities of imagination. It 
would perhaps not be easy to give a brief direct refutation of this, but let us look at 
its implications. Nearly 100 years ago Philip Gosse, in order to reconcile the facts of 
geology with the Hebrew scriptures, advanced the theory that, in his son's words, 
"there had been no gradual modification of the surface of the earth, or slow develop- 
ment of organic forms, but that when the catastrophic act of creation took place, the 
world presented, instantly, the structural appearance of a planet on which life has 
long existed".* The beginning of the universe on this theory occurred some 6000 years 
ago. There is no question that the theory is free from self-contradiction and is 
consistent with all the facts of experience we have to explain ; it certainly does not 
multiply hypotheses beyond necessity since it invokes only one ; and it is evidently 
beyond refutation by future experience. If, then, we are to ask of our concepts 
nothing more than that they shall correlate our present experience economically, 
we must accept it in preference to any other. Nevertheless, it is doubtful if a single 
person does so. It would be a good discipline for those who reject it to express clearly 
their reasons for such a judgment, but the matter is raised here merely to show that 
we cannot grant the inaccessible past freedom to sow its wild oats as it pleases pro- 
vided that it takes care to observe the proprieties when experience begins ; it must 
to some extent conform to the pattern on which we organize the present behaviour 
of the universe. 



Modern Cosmology and the Theologians 

M. Davidson 

The College of St. Mark, Audley End, Essex, England 



1. Statement of the Problem 

Some recent pronouncements on cosmology have left a number of theologians in a 
state of bewilderment and perplexity. Many of them know well that a restatement of 
traditional doctrines which rest on a discredited cosmology would lead to a serious 
disturbance in the faith of a large number of people, and for this reason they often 
maintain a discreet silence. Some are disposed to adopt a suspicious attitude towards 
theories which apparently undermine faith in the Christian dogmas, but they welcome 
pronouncements from men of science which seem to support such dogmas. It is 
suggested that, while discredited cosmologies should not be used to bolster up the 
Christian Faith, nevertheless theologians should not hastily accept the latest pro- 
nouncements of cosmologists as the final word on the subject and should exercise 
great caution before accepting them as necessarily supporting, or proving destructive 



* Father and Son, Edmund Gosse, Chapter V. 



M. Davidson 167 

to, religion. An instance is given of the same law — the Second Law of Thermo- 
dynamics — leading to diverse theological conclusions, and the inference is that 
theologians may find it necessary to adopt a more independent attitude towards 
cosmology. The same remark applies to other sciences but these are not considered 
in this discussion. 

A short summary is given of the views of some cosmologists on theological questions, 
but a caveat on this matter is considered advisable and there follows a quotation from 
a book by the Astronomer Royal as a warning that astronomers are not necessarily 
better equipped than others to make pronouncements on the purpose behind the 
universe. Towards the end a reminder is given to men of science that their explana- 
tions are not necessarily exhaustive and that there may be many unmanifested 
properties in nature. Further, just as the physicist can be said to make use of 
"myths"* — using the word with a more comprehensive meaning than usual — which 
serve a useful purpose, so there may be some justification for myths in theology, for 
example, the myth of the Fall which attempts to explain the origin of evil in the world. 

Dr. Inge (1932) has pointed out that there are three positions which the Church 
may adopt in dealing with astronomical development. The first is to condemn 
astronomical science as impious — an attitude which no one except an extreme 
Fundamentalist is likely to adopt. The second is to admit that traditional doctrines 
do not belong to the natural world with which science deals, but this does not 
necessarily deny that they may possess a higher truth outside the reach of science 
and hence may be regarded as symbolic of eternal truths. The third is to recognize 
the necessity for recasting all theological doctrines which rest on the geocentric 
theory of the universe. Perhaps it was unfortunate that Dr. Inge did not develop a 
scheme for recasting these theological doctrines, though he expressed the opinion that 
anything was better than trying to conceal an open sore which destroys our joy and 
peace in believing. A complete recasting could, however, and possibly would, lead 
to devastating effects on the faith of many earnest Christians. 

Many to-day accept theological doctrines which apparently rest on a discredited 
cosmology and it is equally true that many others have lost interest in theological 
cosmologies, chiefly on the grounds of their naive anthropocentrism and — partly as 
a consequence of this — of their anthropomorphism. It is remarkable that the 
anthropocentric attitude still prevails amongst some eminent men of science whom 
we should truthfully describe as deeply religious, and they experience less disturbance 
to their faith than do many theologians whose lives have been devoted to a study of 
some of the deeper problems of life. Nevertheless, theologians are confronted by 
serious difficulties and we shall now turn our attention in particular to those that arise 
in consequence of the impact on theology of developments in cosmology. 

2. Necessity for Independence of Views amongst Theologians 

In 1948 a paper with the title "Science and Christian Modernism" was read at the 
Modern Churchmen's Conference at Oxford and a brief reference will now be made to 
a few of the points discussed in this paper (Davidson, 1948). 

It was emphasized that the hasty acceptance of scientific theories and using them 
ot support cherished theological views could be detrimental to theology itself. On 
the other hand, the rejection of scientific theories merely on the grounds that they 



* It is not stretching the meaning of the word too far if it is accepted that mythology is an attempt to account for facts 
in the natural order, and that it is more like primitive philosophy than primitive science. 



168 Modern cosmology and the theologians 

seemed to be opposed to certain tenets of the Church (the word "Church" is used in 
its most comprehensive sense) could be even more disastrous. Within comparatively 
recent times certain pronouncements were made on problems of indeterminancy, 
freedom of the will, etc., in the light of the developments in atomic physics, which 
were eagerly welcomed by some theologians — not always to the advantage of theology 
— but as this paper is restricted to problems of cosmology these will now be considered. 

In the paper previously mentioned an example was taken from the law of the 
increase of entropy on which some Christian apologists have been disposed to build 
an imposing edifice, and it was pointed out that some day this law might be shown to 
be valid only under local and restricted conditions and inapplicable in dealing with 
the universe. If this should happen — and there is the possibility that it may, as 
shown by R. C. Tolman (1934) — the structure erected on this foundation would be 
like the house in the parable that was built on the sand. The conclusion from this 
example was, that while the Church should not shrink from presenting the Christian 
Faith in terms of contemporary thought, at least as far as it is competent to do so, 
nevertheless it was essential that it should preserve a certain amount of indepen- 
dence of opinion. It is very unwise for theologians to hang on to the mantles of men 
of science merely because some of their pronouncements seem to support certain 
cherished theological tenets. In addition, diametrically opposite views in the realm 
of theology have been deduced from the acceptance of scientific theories, and this 
will now be shown to have occurred on the basis of the Second Law of Thermo- 
dynamics. 

In a recent work by C. F. von Weizsacker (1951), we are informed that the 
theory of evolution and the Second Law of Thermodynamics were put forward about 
the middle of last century as purely scientific theories, and the former quickly became 
the battle cry of every modern mind. On the other hand, the Second Law of Thermo- 
dynamics remained a technical detail of physics, and subterfuge was used to evade its 
application to the world as a whole. The reason given for this subterfuge is interesting 
— "For the prospect of the heat death of the world, however far off in the future, 
would have shaken the faith that life has meaning". 

It is remarkable that, while this gloomy prospect was disturbing to the faith of 
some, it has had exactly the opposite effect with others as the following instance in 
recent times will show. 

About ten years ago Sir Edmund T. Whittaker (1952), dealing with the law of 
the increase of entropy, made the following statements : "The knowledge that the 
world has been created in time, and will ultimately die, is of primary importance for 
metaphysics and theology ; for it implies that God is not Nature, and Nature is not 
God ; and thus we reject every form of pantheism, the philosophy which identifies 
the Creator with creation, and pictures him as coming into being in the self-unfolding 
or evolution of the material universe. For if God were bound up in the world, it 
would be necessary for God to be born and to perish". 

Later on Whittaker speaks about the evolutionary process in which the indivi- 
dual counts for nothing amongst the lower forms of life, whereas in humanity the 
race loses its value and the individual acquires the supreme value. Nevertheless, 
though mankind and all his works must vanish away (and everyone will agree that 
this must happen if we accept the validity of the Second Law of Thermodynamics), 
yet something remains : "The goal of the entire process of evolution, the justification 
of creation, is the existence of human personality : of all that is in the universe, this 



M. Davidson 169 

alone is final and has abiding significance : and we believe that this has been granted, 
in the eternal purpose of God, in order that the individual man, born into the new 
creation of the Church, shall know, serve, and love Him for ever". 

Here we have two almost antithetical views emanating from the same law; the 
first offers little inducement to mankind to survive; the second, anthropocentric 
though it may seem, does supply a motive for persevering against the most^formidable 
obstacles, and if accepted, would go a long way towards dispelling the gloom and 
pessimism which overshadow the world to-day. Without offering any opinion on the 
validity of either argument, one may merely add that it would serve no useful pur- 
pose to disturb the faith of those who believe that there is purpose in the universe and 
an ultimate goal for mankind. The two examples mentioned above (and others 
could be given) should be a warning to theologians. They should beware of accepting 
the latest pronouncements of cosmologists as final and of accepting them for destruc- 
tive or constructive purposes, and equally they should avoid all manifestations of 
undue alarm at such pronouncements. That this last warning is not out of place may 
be shown by the following instance. 

At the Modern Churchmen's Conference at Cambridge in 1950 a number of the 
members were very disturbed by some recent pronouncements of a cosmologist who 
had expressed the opinion that Christianity offers merely an eternity of frustration. 
Their main anxiety was due to their dread that the faith of many might be under- 
mined by such a statement. The writer of this article, who did not share their view 
in the matter, endeavoured to reassure them (not, it is to be feared, with much 
success). It is very difficult to understand why theologians should be perturbed by a 
personal view of this kind which, after all, cannot be regarded as authoritative. 
Incidentally, it may be remarked that the Founder of Christianity warned His 
followers that frustration — temporary at any rate — would be their lot, and frustra- 
tion is not necessarily an evil thing. That the cosmology of the book (Hoyle, 1950) 
is very interesting, no one will deny, nor will anyone deny that previous cosmologies 
have also been most interesting ; but the final words on the subject have not yet been 
spoken. Regarding the pronouncements of cosmologists on some of the deeper 
issues of life, the following quotation from the end of a work by the Astronomer 
Royal, Sir Harold Spencer Jones (1952), is relevant : "The task of the astronomer 
is to learn what he can about the universe as he finds it. To endeavour to understand 
the purpose behind it and to explain why the universe is built as it is, rather than on 
some different pattern which might have accorded better with our expectations, is a 
more difficult task ; for this the astronomer is no better qualified than anyone else". 

These words should be a standing rebuke to all cosmologists who venture to make 
pronouncements in realms in which they have no special qualifications to do so. 

3. Recent Cosmological Pronouncements 
We shall now consider the most recent views of a well-known cosmologist, though 
it must be admitted that many of his theories have not met with general approval. 
The late Prof. E. A. Milne, who wrote extensively on the subject of cosmology, was 
a deeply religious man, and in one of his works which appeared nearly twenty years 
ago, Milne (1935) claimed that, in the view of the universe that he advocated, he 
could say he had found God. "For the universe seems to be a perfect expression of 
those extra-temporal, extra-spatial attributes we should like to associate with the 
nature of God". Later he admits that the physicists and cosmologists need God 



170 Modern cosmology and the theologians 

only once, to ensure creation, but in a recent — and posthumous — work Milne is more 
definite, and a short summary of his views expressed in this book (Milne, 1952) now 
follows. 

The main thesis of the book is that God formed the visible universe, that it bears 
the marks of a created article, and that He has also created the laws of nature. 
These are not special ad hoc creations but, following the view of Ernst Mach, the laws 
of nature are a consequence of the structure of the universe. Milne found evidence 
of the creation of the universe by a rational Creator from the rationality of the 
universe itself. We are able to know something about the rules that nature obeys 
from the study of our own rational thought-processes which inform us of the limita- 
tions placed on nature if it is to be rational. He rejects the hypothesis of Bondi and 
Gold (1948) (see also Bondi, 1952) on continuous creation of matter which, he says, 
is irrational "because it requires the specification of a rate of creation of matter per 
unit volume of space". He thinks that no reason can be given for the choice by 
God of any particular value of this quantity, but asserts that his own hypothesis 
that the universe was created as a point-singularity is not open to such objection. 

Many will probably think that this argument is by no means convincing and they 
will not feel that much assistance is afforded by Milne's definition of "rational", 
which is as follows : "We say that the universe is rational when laws of nature, 
predicted a priori, are found observationally and experimentally to be obeyed in 
nature". This definition is largely based on Milne's view that science should aim at 
becoming less empirical and more deductive, but even if it is admitted that physics 
should become more deductive, we cannot but ask how far the deductive method is 
applicable to biology. Obviously Milne foresaw a difficulty here because he admits 
that he has not excluded the possibility of divine interference in the details of 
biological evolution. Readers will probably agree that a serious difficulty arises 
once such divine interference is admitted, because the deductive method is then 
inapplicable unless it is capable of predicting the minute details of the mind of the 
Creator. Limits of space prevent an exhaustive examination of this book but one 
other point will be referred to. 

In the last chapter Milne regards the infinity of the galaxies as an infinite number 
of scenes of experiment in biological evolution, but he is perplexed by the difficulty 
felt by many Christians in connection with the infinite number of planets that have 
evolved from the infinite mass brought into existence at the epoch t — 0. He even 
discusses the question of the incarnation — whether it was a unique event on the 
Earth or whether it has been re-enacted on each of a countless number of planets. 
He gives reasons — not very convincing from the theological point of view — for 
favouring the former view and suggests that some day we may be able to signal the 
news to other planets. He thinks it is not outside the bounds of possibility that radio - 
signals apparently proceeding from the Milky Way are genuine signals from intelli- 
gent beings from other planets, and that "in principle, in the unending future vistas 
of time, communication may be set up with these distant beings". 

There are, of course, other cosmologies, but as they do not deal so fully with the 
theological implications as Milne's, they do not lie within the sphere of the present 
discussion. As a personal opinion only, which must not be regarded as expressing 
the views of theologians in general, it might on the whole be better if cosmologists 
confined themselves to their own particular subject and left the philosopher and the 
theologian to draw their own conclusions. 



M. Davidson 171 

4. Suggestions on a Constructive Police 

Can anything constructive be built on the results of modern research in physics 
and cosmology (it is impossible to separate the two in such an enquiry) ? First of all, 
physicists and cosmologists should realize — as indeed most of them do — that their 
scientific explanations are by no means exhaustive, and while they can observe such 
things as masses, lengths, times, etc., there may be an almost unlimited number of 
other properties which are not only unmanifested but which, for the greater number 
at least, will ever remain so. Then they may be prepared to give some sympathetic 
consideration to certain religious tenets — even to those that are based on mere 
mythology which finds a place in the Christian religion. Physicists themselves are 
not averse to the use of "myths" — if the word may be used to signify the basis of some 
of their theories. It is only necessary to mention such myths as the ether of space 
which, with all the extraordinary properties with which it was endowed, once played 
an important role. Are the "solar system" -atoms, the various fundamental particles, 
the differential equations of wave mechanics, and so on, anything more than myths 
used to describe a number of puzzling phenomena ? They are not permanent myths 
but serve their purpose until some less crude myths supplant them. In the same way, 
is the theologian not justified in using the myth of the Fall of Man to explain the fact 
of evil in the world — a fact which few men of science will deny ? Admittedly it is a 
crude explanation, but there is a deep moral significance in it all which even biological 
evolution has not yet destroyed. Other similar instances of the use of mythology 
or of legendary lore could be cited but sufficient has been said to suggest that men of 
science and theologians, each in their own sphere, can come more and more to see 
each other's point of view. This is all the more necessary amid the perplexities of the 
present time when the eternal questions, Whence? Whither? Wherefore? still 
remain unanswered. 

5. Some Quotations 

In conclusion, the following quotations from men of science and philosophers, are 
relevant to certain points discussed in this chapter. 

"Science is a certain kind of knowledge, notable for its high degree of reliability 
within its self-imposed limitations ; so it must necessarily contribute to philosophy 
in the wide sense in which we have used the word. Religion is, in part, also a kind of 
knowledge, relating but little to things, and more to men — both being considered in 
their relation to God". (Taylor, 1951.) 

"Philosophy and science do not answer our deepest questions, nor do they 
solve effectively our most pressing problem, that of human conduct". (Taylor, 
1951.) 

"Examples of legitimate philosophies are numerous. One thinks, for example, 
of various systems of theology for which men have been ready to die, yet 
which, with larger experience, have been discarded. Their holders have held 
them indispensible for rationalizing their own limited religious experience, and it 
would have been the height of folly for them to have exchanged such theologies for 
a not yet achieved inclusion of religious experience within the scientific scheme". 
(Dingle, 1952.) 

"I cannot, however, help feeling it is likely to be more important for religion in the 
future to have a theology that is founded on the reality of religious experience, than 
to have one that builds its doctrines upon supposed events in the past : supposed 



172 



Modern cosmology and the theologians 



events which some of the best scholars of history are unable to regard as established 
beyond doubt by the rules of evidence accepted in other fields of historical research". 
(Hardy, 1951.)* 



References 

Bondi, H 1952 

Bondi, H. and Gold, T 1948 

Davidson, M 1948 

Dingle, H 1952 

Hardy, A. C 1951 

Hoyle, F 1950 

Inge, W. R 1932 

Milne, E. A 1935 

1952 

Spencer Jones, H 1952 

Taylor, S 1951 

Tolman, R. C 1934 

Weizsacker, C. F. von 1951 

Whittaker, E. T 1942 



Cosmology (Cambridge University Press). 

M.N., 108, 252. 

The Modern Churchman, XXXVIII (Sep- 
tember 3rd) (Blackwell, Oxford). 

The Scientific Adventure (Pitman, Lon- 
don). 

Science and the Quest for God (Lindsey 
Press, London). 

The Nature of the Universe (Blackwell, 
Oxford). 

The Church and the World (Longmans, 
London). 

Relativity, Gravitation, and World Struc- 
ture (Oxford University Press). 

Modern Cosmology and the Christian Idea 
of God (Oxford University Press). 

Life on Other Worlds (English Univer- 
sities Press, London). 

Man and Matter, Essays Scientific and 
Christian (Chapman and Hall, London). 

Relativity, Thermodynamics, and Cos- 
mology (Oxford University Press). 

The History of Nature (Routledge and 
Kegan Paul, London). 

The Beginning and End of the World 
(Oxford University Press). 



* A few weeks after the manuscript of this paper had been sent to the Editor, an article by Rudolf Bttltmann, the sub- 
stance of his broadcast, "What is Demythologizing?'*, appeared in The Listener of 7th February, 1953, which has some points 
similar to this article. This similarity, however, is entirely accidental, and the writer of this paper has not altered his original 
draft in consequence of Prof. Btjltmann'S interesting broadcast which he has since read, but which he did not hear. 



SECTION 3 



DYNAMICS 



"Nous devons done envisager l'etat present de PUnivers 
comme l'effet de son etat anterieur, et comme la cause 
de celui qui va suivre. Une intelligence qui, pour un 
instant donne\ connaitrait toutes les forces dont la 
nature est anim^e, et la situation respective des etres 
qui la composent, si d'ailleurs elle 6tait assez vaste 
pour soumettre ces donn^es a 1 'analyse, embrasserait 
dans la meme formule les mouvements des plus grands 
corps de l'Univers et ceux du plus 16ger atome; rien 
ne serait incertain pour elle, et l'avenir comme le passed 
seraient presents a ses yeux. L, 'esprit humain offre, 
dans la perfection qu'il a su dormer a l'astronomie, une 
faible esquisse de cette intelligence". 

Pierre-Simon Laplace, Essai Philosophique sur les 
Probdbilites, Paris, 1814. 



Vistas in Celestial Mechanics 

D. H. Sadler 

H.M. Nautical Almanac Office, Royal Greenwich Observatory, 
Herstmonceux Castle, Hailsham, Sussex, England 

Summary 

A brief review of the present status of celestial mechanics, in relation both to the need for increased 

accuracy and to the new methods made possible by the use of electronic digital computing machines. 



1. Problems 
The aim of that branch of astronomy known as celestial mechanics is, in its most 
general form, the determination of the motions of celestial bodies under their mutual 
gravitational attraction and under such other forces as may exist. The solar system 
has provided so far a sufficient number of problems, as well as the observations 
necessary for their numerical solution and subsequent verification. Most of these 
practical problems have been solved to a high degree of accuracy, enabling the 
positions of the Sun, Moon, planets, satellites, and comets to be predicted for many 
years in advance. But these solutions, although generally of the same or superior 
order of accuracy to the observations with which they are to be compared, are not 
entirely satisfactory. 

There is no theoretical or literal solution to the w-body problem, or at least none 
admitting of practical development. The classical solution consists of the expression 
of the co-ordinates of the celestial bodies as trigonometrical series, with numerical 
coefficients, and with time as the independent variable ; even so it has been applied 
rigorously only in the case of the Moon-Earth-Sun system and, in most other cases, 
the time is allowed to appear as a factor outside the trigonometrical expressions, so 
limiting the period of validity. Such solutions represent the culmination of mathema- 
tical and analytical skill developed over more than a century's effort by some of the 
world's foremost mathematicians; but the amount of labour involved, both 
theoretical and computational, is enormous and an extension of the accuracy, by 
these classical methods, is quite prohibitive in computational effort. 

Thus the state of the subject before the second World War was an uneasy balance. 
On the one hand the immediate requirements of the practical astronomer had been 
met and there was little inducement for new work in a subject in which few new 
lines of attack were possible ; on the other hand the unsatisfactory nature of the 
solutions, and their unsuitability for further extension, was recognized by the few 
workers in the field. 

The great technological progress during the war has led to two, or possibly three, 
developments that have transformed the whole subject of celestial mechanics, in 
respect of both demands of accuracy and methodology. Firstly, the development of 
quartz crystal (and other) clocks and the possibility of improved observational 
techniques due to electronic observing and recording gave warning that the present 
standards of accuracy in the fundamental ephemerides of the Sun, Moon, and planets 
might soon be inadequate. Simultaneously, the development of electronic automatic 

175 



176 Vistas in celestial mechanics 

digital computing machines not only provided the tool by which the required exten- 
sion in accuracy could be made possible, but also gave rise to a new method of 
approach to the classical problems. The third development is in a very different 
field, namely that of "space -travel" ; little serious work has been done on the many 
problems of celestial mechanics that will arise in the "navigation" of a "space-ship", 
capable of supplying a limited motive force to itself, in the gravitational field of the 
solar system. It is, however, probable that the practical problems will have to be 
solved inanad! hoc fashion, and it is certain that celestial mechanics will be able to 
supply all necessary solutions to the general problems concerned whenever the 
technological difficulties of rocket-travel have been overcome. 

2. Ephbmbris Time 

The first and most important practical problem to be solved by the aid of celestial 
mechanics is the accurate determination of time. Clocks can keep time over short 
periods more accurately than astronomical observations can be made, but the 
latter still provide the only means over long periods. The rotation of the Earth can 
no longer be regarded as uniform, being subject to unpredictable erratic changes in 
rate in addition to the gradual retardation due to tidal friction and to small seasonal 
fluctuations. Yet day-by-day determinations of time must be made by observation 
of the directions of stars, relative to the rotating Earth ; time so determined is 
Universal Time (U.T.), and is not uniform. A new time, Ephemeris Time (E.T.), 
defined by means of the revolution of the Earth round the Sun, has accordingly 
been introduced* and this is uniform, to the best of our present knowledge. 

Ephemeris Time, being uniform, can be chosen so that its origin and rate can be 
identified with the independent variable ("time") of the theories, tables and ephemer- 
ides of the Sun, Moon, and planets. Actually it is in practice defined by reference to 
Newcomb's tables of the Sun, and the ephemeris of the Moon is modified! to bring 
both ephemerides into accord with this common argument. Ephemeris Time can be 
determined observationally by comparison of the positions of the Sun, Moon, planets 
(and satellites) in their orbits with those predicted by gravitational theory, without 
reference to the rotation of the Earth. Naturally, its value at any particular instant 
cannot be found by such observations to the high accuracy associated with Universal 
Time, but the average value of the difference between the two times can be obtained 
accurately over a period of, say, one year, and, with the better observing techniques 
now envisaged, perhaps in as little as a month. Owing to its rapid motion and in 
spite of the difficulty of accurate observation, the Moon offers the best means for the 
practical determination of E.T. by observation, and a precise ephemeris of the Moon 
is thus required. 

3. Ephemeris of the Moon 

A very accurate knowledge of the Moon's position and motion is also required for 
the determination of geodetic distances on the Earth's surface by means of observa- 
tions of lunar occultations and solar eclipses. 

This requirement is being met by the direct calculation of the lunar ephemeris 
from the numerical trigonometrical series developed by Brown (1919), thus avoiding 



* Recommendation No. 6 of the Paris Conference on the Fundamental Constants of Astronomy, Bulletin Astronomique, 
tome XV, fascicule 4, p. 291 (1950). Adopted by the I.A.U. as Resolution No. 6 of Commissions 4 and 4a. 

t Recommendation No. 5 of the Paris Conference (also Resolution No. 2 of Commissions 4 and 4a). See Sadler (1951). 



D. H. Sadler 177 

the approximations necessarily involved in the use of the Tables. The synthesis of 
some 1500 separate terms of the series would have been quite impracticable without 
the aid of electronic computing machines, and is a considerable task even with such 
machines. The ephemeris, to 0-001 in R.A. and 0"01 in Dec, is being computed 
from 1952 onwards, but will not appear in the almanacs in its proper place until 1960. 

An artificial satellite of high albedo and of stellar appearance, revolving round the 
Earth in a period of a few hours, would serve as the hand of a clock of far greater 
long-term precision than anything available to-day. Such a body, even if revolving 
round the Earth in a stable orbit without self-generated forces, would provide 
problems in celestial mechanics of considerable difficulty if its position were to be 
calculated to an accuracy sufficient for the determination of time. It is possible that 
such artificial satellites will eventually be launched, but it is certain that the require- 
ments of direct observation will be a secondary consideration ; both observation and 
the computing problem will thereby be made more difficult. 

One of the fundamental problems of celestial mechanics is to determine the 
motions, for all time, of a number of bodies of known mass moving under their mutual 
gravitational attraction, given at any one instant their positions and velocities in 
space. The very simplicity of its statement emphasizes its generality and its difficulty ; 
it has attracted the attention of many eminent mathematicians, but, in spite of 
great mathematical achievements and ingenuity, no solution has been found which 
does not involve almost prohibitive numerical computation. Two entirely different 
methods have been used — those of "general perturbations" and of "special 
perturbations". 

In the first of these methods the co-ordinates of the bodies are expressed as 
trigonometrical series with time as the independent variable. The series converge 
slowly even in the more favourable cases, while the possibility of a close approach of 
two of the bodies will usually make the method impracticable. But the availability 
of computing machines, which can be made, for instance, to multiply trigonometrical 
series of some hundreds of terms, is changing the significance of a given level of 
impracticability. In fact consideration is being given by Broitwer, Clbmence, and 
Eckert, in the United States, to a comprehensive numerical solution of the motions 
of all the planets in the solar system. The great advantage of this classical method is 
that the positions of the bodies, consistent with the adopted masses, can be obtained, 
relatively simply, for any time merely by substituting that time in the trigonometrical 
series. 

4. Special Perturbations 

The alternative method consists of the direct numerical integration, by step-by- 
step methods, of the differential equations defining the motions of the bodies con- 
cerned. It has the great merit of simplicity of principle and involves no mathematical 
analysis ; but it suffers from the disadvantage inseparable from numerical work of 
this nature — namely the accumulation of building-up errors. This means that no 
solution can be valid, to the required precision, for more than a limited time ; the 
number of significant figures required to be kept in the integrations increases too 
rapidly with the number of steps. Until the advent of the automatic computing 
machines, the method had been limited to various special problems generally in 
which the positions of the perturbing bodies were assumed known, so that only 
three second-order equations had to be solved. The position is very different now : 

13 



178 



Vislii* iii .rlcstial HU'i'!i4iriicH 



this method has been used by BROUW3SR, OuMEEKm, and Kckkkt (1951) on tlie 
I.B.M. Selective Sequence Electronic (Calculator to compute the positions of the 
five outer planets (Jupiter to Plulo) for 1 he period Hi;*);} 2()(i0, with such success that 
the results have been adopted by the International Astronomical Union m the basis 
for the fundamental ephemcrides from I Wit) onwards. Xaik (li)ol) has demonstrated 




Fig. I. A p-iu*nil virw of the Select ivo Soquenoe Hli-etroiiif Calculator {SSlCf) of the 
International ItusiiK'sw Machines Corporation, by whose permission this photograph is 
reprodueeiL Tliis machine was used for the ctik-idatioit, by numerical integration, pf the 
fundamental (>|)hi'iin Tides of the five outer planets, and Cor tlie direct calculation of the 

Iiimili- ephi'meris from Uko\vn*S theory an modified according to ihe resolution iidoptt-d 
at tlie Home moating, 1 !)">:!. of the LA. I*. 



the power of K I )S A( ' (an automatic computing machine in Cambridge) in computing 
the very precise orbits of minor planets required as fundamental standards of 
reference for right ascensions. .Many other integrations {Jupiter's outer satellites, 
Icarus and other special minor planets), now being laboriously done by hand, will 
doubtless soon be done on these automatic machines (Fig. I and Fig. 2). 

The method of special perturbations thus now offers a practical solution to many 
of the problems of celestial mechanics ; but it can never replace entirely the classical 



D. H. BADIiBB 



I7!l 



method, since it fails explicitly to exhibit the dependence of the perturbations on the 
masses ami periods of the perturbing bodies. The principal inm.s of t he expansions 
of the classical method are still essential to the full understanding of the physical 
(and mathematical) basis of the theory, .Moreover, the short time-interval necessary 
for the inner planets makes numerical integration unsuitable for fundamental 
orbits, for which an essential condition is consistency and continuity over the whole 
period of observation, A new theory of Mars has accordingly been undertaken by 




Fig, 2. A general view of tlu> BDSAC (Electronic Uolay SLowifjf Automatic Calculator) 
jit tin" University Mathemattoa] Laboratory, f'ariibridyw, by wlui.se permission tkis photo* 
praph is reproduced. A programme lias* been worked out mid used ciri this tiwliini* for com- 
i ul !!:■_' the orbit of a iriimn- plitiii-i with hijjh precision under the attractions nf the Sun and 

f '.-! 'tiirtiin-* planets 



Clemestck (19+9), and the first -order theory has already been published. In this 
theory, which shows up some significant errors in Newcomb's theory, all periodic 
terms are included whose coefficients are as large as 0*0001; it must surely be the 
most carefully planned and accurately executed planetary theory ever undertaken, 
and could hardly have been possible with the calculating equipment available in 
Xkwcomh's day. Good progress is being made with the second-order theory, for 
which the published work furnishes such a sound basis ; it is already clear that the 
perturbations of the second order will introduce further substantial corrections to the 
present theory. 

No reference has so far been made to the by-no-means inconsiderable theoretical 
work that is in progress in celestial mechanics. Although in practice numerical 
answers are nearly always required, there are many questions to which the final 
ephemeris does not give the answer directly — for instance, secular perturbations of 
the elements of the <u bit . It is a measure of the rising interest in the subject, fostered 
largely by the "school" in the United States and now actively encouraged by the 



180 An introduction to the Eclipse Moon 

ability to undertake computations that were previously impracticable, that no less 
than three major textbooks on celestial mechanics are being written, after a gap of 
nearly thirty-five years ! 

After being in a state of suspended progress for many years, celestial mechanics 
thus is faced simultaneously with new problems and with the mechanical means of 
solving them. There is no doubt that the theory will keep pace with the demand and 
that, even in the difficult field of man-made bodies, it will continue to provide, as it 
must, solutions to an accuracy greater than that of observation. 



References 

Brown, E. W 1919 Tables of the Motion of the Moon, New Haven. 

Clemence, G. M 1949 Astronomical Papers Prepared for the Use of the 

American Ephemeris and Nautical Almanac, 
Vol. XI, Part II. 

Eckert, W. J.; Broxjwer, Dirk; and 

Clemence, G. M 1951 Ibid., Vol. XII. See also the review in Observa- 

tory, 1952, 72, 35. 

Naur, P 1951 M.N., 111, 609. 

Sadler, D. H 1951 M.N., 111, 624. 



An Introduction to the Eclipse Moon 

R. d'E. Atkinson 

Royal Observatory, Greenwich, London, S.E.10 

Summary 

Observations of the Moon's place by different methods differ in such important systematic ways that it is 
advisable in practice to think of the Earth as accompanied by a number of different, but incomplete, 
Moons, whose positions, though nearly identical, are in many ways logically independent. The seven 
most important of these Moons are briefly described, and the various methods which have been used to 
study the place (and diameter) of the "Eclipse Moon" are summarized, including a method devised by 
the author and successfully tried out in 1948 and 1952. Either a total or an annular eclipse can now 
give the (Eclipse) Moon's place more accurately than a year's occultations can give that of the Occultation 
Moon; the important problem of fusing these two (and others) of the aggregate of Moons is briefly 
touched on. 



The Nautical Almanac will shortly start giving the Moon's place to a hundredth of a 
second of arc, and it already gives the semidiameter and parallax to this accuracy. 
The precision of individual observations is, of course, considerably less ; but it is not 
at all the case that the observed values differ from the computed ones by amounts 
which are simply random. The Moon's outline is full of features which vary systema- 
tically, and it departs fairly often by more than two full seconds from the best 
circular arc that can be fitted to it at any given moment ; there is also evidence that 
the best semicircles for the east and west limbs have slightly different radii (and 
perhaps also different centres), and that these differences may change as the libration 
changes [1]. The libration does in fact change rapidly in a two-dimensional pattern, 
with two incommensurable periods, and over a range large enough, in each co-ordinate, 



R. d'E. Atkinson 181 

to sink even the largest mountain entirely below the limb ; thus the apparent place 
inferred for the Moon from any given observation will depend greatly on the libra- 
tional situation at the moment, and on the point or points actually observed, but also 
on the extent to which the particular arc of a circle, to which those points could in 
principle be reduced by applying limb-corrections, differs from the complete circle 
that may be considered characteristic of the mean whole Moon. In practice, limb- 
corrections hardly ever are applied, as yet, to any observations of position (except in 
eclipses); moreover, different methods of observation are necessarily subject to 
different systematic errors ; and since in addition they give systematically different 
weights to different parts of the limb at different times, including zero weight for 
large lengths of it during large (and systematically different) fractions of each month, 
it really is the case in principle that the earth is at present accompanied not by one 
Moon, but by a sort of out-of-focus aggregate of several incomplete Moons, which 
are logically distinct in the sense that a correction to the place of one of them is not 
bound to apply also to another. Any large correction, of course, such as that due to 
the E.T.-U.T. difference, can approximately be applied to all, and indeed no predicted 
place of the "Eclipse Moon" is ever based on previous eclipse observations at all, but 
only on observations of some other member of the aggregate ; systematic disagree- 
ments with the places observed in eclipses are thus only to be expected, but reliable 
data are still somewhat scanty for assessing their actual values. The purpose of the 
present note is first to summarize the properties of these different Moons, and then to 
review briefly the observations which have dealt directly with the Eclipse Moon; 
it will appear that this is in many ways peculiarly isolated from all the others. 

(1) The Moon of Brown's Tables 

If the Empirical Term is removed, this Moon's place is based on pure gravitational 
theory; the disposable constants of the equations have been adjusted to suit long 
series of past observations as well as possible, but even so this Moon differs systema- 
tically from all the observational ones in two important respects : the place is given 
in terms of a strictly uniform time (E.T.) instead of a time involving the earth's 
rotation (U.T.) ; and the mean (orbital) latitude is necessarily zero, whereas the mean 
latitudes of the observational Moons are generally slightly negative. The latter fact 
may imply that the centre of apparent figure lies slightly south of the centre of 
gravity, presumably owing to the mountainous nature of the south polar regions ; 
from the former consideration it follows [2] that the comparison of "Brown's Moon" 
with any observational one gives an indication of the difference between the two 
kinds of time. Brown's Moon has sometimes been supplemented by direct integra- 
tion ("special perturbations") if its behaviour over a short interval was required 
with extreme accuracy, e.g. in an eclipse [3]. 

(2) The Moon of Photographic Limb-Surveys 

(See Hayn [4], Watts [9].) This Moon's place is not observed at all, and its diameter 
is doubtful ; but an accurate knowledge of its other characteristics is essential if any 
real improvement in the place and diameter of any other Moon is to be made. Only 
the bright limb exists, of course; and in fact even at "full" Moon there is usually 
"defective illumination" over a large fraction of one semicircle. Defective illumina- 
tion is serious, not only because the (mean) terminator then lies a little inside the 
(mean) limb, but because the terminator is the limb as seen from the Sun, and the 



182 An introduction to the Eclipse Moon 

libration (and thus the limb-contour) differs, as between a solar and a terrestrial 
observer, by quite large amounts even for small extents of the defect. (0"- 1 means 
0-84° of libration-difference.) This Moon exists only during hours of darkness, and 
most observations are thus made within a week of full moon ; but even so it is clear 
that it will seldom be possible to use more than 200° or 220° of limb on any one photo- 
graph, without running into these libration-differences, and the complete 360° 
contour characteristic of any one pair of libration-co-ordinates (I, b) can only be built 
up out of pieces. There are, however, obstacles to this; the absolute radius depends 
on the seeing and transparency and cannot be used to control the fit, and this lack 
can be directly made up only if one obtains substantial overlaps, between successive 
regions of the limb, on photographs taken at the same I and b. In practice, the same 
pair of I, b values does not usually recur again at all closely for a long time [5], so 
that one really has to build up a surface rather than a set of circles. Moreover, the 
complete circular outline for any one pair of I, b values would in any case remain 
incompletely observed at one point ; photographs near full moon give a good overlap 
with both east and west limbs, but (at least as far as most outlines are concerned) 
they do so near one pole only. The "Cassini rules" have the effect that whichever 
pole is tipped towards us at full moon, the limb beyond it is always the defectively 
illuminated one, and the combination of the successive overlapping arcs (whether into 
circles or into a single sphere) cannot therefore be checked by proper closure -tests. 
Watts is at present engaged on a very thorough programme of constructing the 
best mean sphere characteristic of all observed arcs, but this work is necessarily 
very difficult. 

(3) The Heliometer Moon 

This somewhat resembles the former, except that observations are visual, and that 
points on the limb are also related to craters on the surface (usually Mosting A.) 
It is this work which has recently given information on the difference between east 
and west radii, and their dependence on libration ; the mean inclination and node of 
the Moon's equator, and the constants of the "physical" libration, are also inferred 
from it [6]. 

(4) The Moon of the Meridian-circle Observer 

This Moon does not exist within about three days either way from new moon, but it 
can exist in full daylight; its east "limb" (really only one point, at any one transit) 
does not exist during the first half of the month nor its west one during the second ; 
and although both these points, or both the north and south pair, are occasionally 
listed in the Almanac as "observable" at the same transit, one in each pair is always 
(necessarily) shown as "defectively illuminated". Further, since the position-angle 
of the axis cannot exceed about i 25°, there are four regions, each about 40° long, 
which do not exist at all; meridian observations cannot well be corrected for limb- 
features and must be taken as they stand, and thus the "Meridian Moon" may differ 
systematically from all Moons located by means of their complete contours, whether 
or not corrections are applied in their case. (The meridian observer can apply correc- 
tions only if he can remember, and specify accurately from memory, just how he laid 
the wires up against the irregularities with which he found himself suddenly confronted 
at transit.) The diameter of the Meridian Moon is seldom directly observed and is in 



R. d'E. Atkinson 183 

any case falsified by irradiation; the much more extensive data which have been 
obtained for the Sun [7] suggest that it may also be subject to large personality- 
effects, and that these may themselves depend strongly on the zenith distance. 
In contrast to the "Meridian Sun", an error in the diameter usually causes an 
error in the place. On all counts, therefore, there is little hope of discovering the 
precise short-period behaviour of this Moon; but its general behaviour is well 
known. 

(5) The Occultation Moon 

This is the most precisely known Moon nowadays, but it too is very unevenly ob- 
served. It scarcely exists within about three days either way from new moon, 
nor during daylight or bright twilight ; in addition, dark-limb observations are easier, 
and can be made with many more stars, than bright-limb ones, and disappearances 
are generally more reliable than reappearances, and also much more often observed. 
For all practical purposes indeed, the west limb does not exist when bright, and the 
main weight is in the dark-limb disappearances (age 5 to 13 days, say) ; the corrections 
derived for the Moon's place are often meaned by lunations, and these means may 
sometimes therefore be discussed as if the orbital and libration elements for all the 
observations had been what they were at the age of 9 days or thereabouts. The 
values so meaned [8] show oscillations in latitude which it would certainly be 
impossible to explain by oscillations in U.T. ; Watts has shown [9] that the applica- 
tion of individual limb-corrections would reduce them considerably, but they may 
also be due in part to errors in the star-places, which necessarily affect them. The 
Occultation Moon's diameter exists, and is entirely unfalsified by irradiation; it 
cannot be based on the simple comparison of dark-limb observations near first and 
last quarters, since it would then hardly be separable from the parallactic inequality, 
but it has been determined by disappearances and reappearances, and also by pure 
dark-limb disappearances, preferably in regions rich in stars [10]. It follows that 
there is a tendency for the results to be tied to special values of b ; for example, 
occultations in the Pleiades involve b rm — 5|°, and occultations (faint stars) in lunar 
eclipses involve b & [11]. 

(6) The Astrometric Moon 

This hardly exists at all, as yet, but it is likely to become very important. The 
difficulty in getting a place by direct photography, as is regularly done for minor 
planets, has been that the Moon's motion is appreciable even in the short time 
needed to photograph suitable comparison-stars ; the new technique (Maekowitz) 
of countering this motion by means of a slowly-tilting glass plate, covering only the 
Moon's image, seems most promising. What is obtained is a place for the centre of 
the entire arc which is photographed, since although the radius is falsified by irradia- 
tion the centre is unaffected; this is a very great advantage. With the exception of 
the Eclipse Moon, this is the only one for which the apparent places of a really large 
number of limb-points are observed simultaneously, before the librations have had 
time to change, and each photograph can in principle supply its own limb -corrections, 
including a direct and valuable check on their position-angles. Only the bright limb 
exists, of course, but observations just before and after full moon should help to 
show how far the centres of figure of the east and west limbs coincide. As with the 



184 An introduction to the Eclipse Moon 

Occultation Moon, systematic errors in the star-places go straight into the Moon's; 
but their random errors are less serious. 

(7) The (Solar) Eclipse Moon 

This exists only at an age (or elongation) entirely outside the range of all other 
observational Moons, namely zero. Its libration in latitude is always nearly zero also, 
and its contours thus form almost a one-parameter array, being nearly determined 
by the libration in longitude alone. It is the only Moon whose complete contour can 
be observed before the librations have changed. This may obviously be done at an 
annular eclipse, but it can be nearly as well done at a total one too ; if one has cameras 
a little outside the track, on both sides, and takes photographs for about 30-40 min 
near mid-eclipse, the rapid change in position-angle allows the entire limb to be 
covered, with good overlaps. This method was tried at the Sudan eclipse of 25th 
February, 1952, but the seeing in the desert was poor. The place of the Eclipse Moon 
is tied to that of the Sun ; its diameter is separable from the Sun's in principle, but 
usually not in practice. The approximate place is known back to great antiquity, 
and gives some information on the secular change in the Earth's rotation, as well as 
useful checks on early chronology. The following are the principal modern observa- 
tional methods ; regrettably, work has been confined almost entirely to total eclipses, 
and the apogee half of this Moon's orbit is nearly unrepresented. 

(a) From timed position-angles of the line of cusps, in the partial phase, one can 
derive corrections to both co-ordinates of the Moon's place, provided one observes 
long enough to obtain a good change in P.A. The diameter clearly cannot be obtained 
in this way (or at least not without knowledge of the limb-detail), because for two 
circular outlines the line of cusps is perpendicular to the line of centres whatever the 
diameters. If, however, one also measures the separation of the cusps, the diameters 
of both Sun and Moon are in principle obtainable ; in practice, one of the two still has 
usually to be assumed, and in addition there is obviously more risk of an error due to 
irradiation than in the case of position angles. Most early work of this kind was 
done at established observatories, where the eclipse was only partial anyway; some 
of the best visual results were obtained with heliometers, but filar micrometers, and 
even measurements on the two cusps separately, were also successfully employed. 
Some important cases, with the probable errors in right ascension and declination, 
were: Wichmann, 1851, ± 0-3 and ± 0-2 [12]; Kobold, 1890, 1891, and 1893, 
about ± 0-18 in each co-ordinate [13]; also 1900, ± 0-26 and ± 0-28 [14]; 
Merlin, 1905 (using a projected image), ± 0-1 and ± 0-2 [15]; also 1908, ± 0-4 
and ^ 0"5 [16]; Naumann, 1912 (applying limb -corrections to the cusps), i 0"07 
and ± 0-06 [17]; also 1914, ± 0?12 and ± 0-08; and 1927, ± 0-26 and ± 0?17 
[18]. Photographic work was started in 1912, particularly by Hayn, ^ 0"08 and 
± 0-11 [19]; also 1914 [20]; in 1927 [21] he adopted the procedure of determining 
the centres of the lunar and solar limb-arcs from measurements on twenty points of 
each (not using the actual cusps at all), and he also measured the distances and posi- 
tion angles of each cusp from the Sun's centre, so determined; his probable errors 
were about i 0''07. 

The above results were among those which claimed the smallest probable errors 
[22], but systematic errors (e.g. in timing) may be quite as large; timing was in fact 
rather doubtful in some cases. And the whole theory of the method breaks down if 



R. d'E. Atkinson 185 

the Moon is elliptical (unless limb-corrections based on a circle are applied) or if its 
two halves are corrected to circular arcs whose centres and radii are not identical, 
since one cusp is pretty sure to be in the eastern half and one in the western. 

(b) Cinematography was first used at the "limiting" eclipse of 1912, when it was 
devoted mainly to deciding whether the eclipse was total or not, and where the 
centre-line ran. Several observers set up one or more cine cameras near the centre- 
line [23], and decided where the line had been, and when mid-eclipse had occurred, by 
mere inspection, picking those photographs which showed the most uniform inten- 
sities of three or four Baily's beads that were well distributed round the Moon. 
This method is very seldom applicable ; it is not really objective ; and it obtains the 
place of a Moon which is different from all those discussed above, namely the Moon 
of beads due to the deepest valleys only. (The "centre of gravity" of a bead is an 
uncertain distance above the valley floor, even if the latter is known.) 

(c) A much better cinematographic method was devised by Banachiewicz, and 
was tried out, under his direction, by Kordylewsky at the 1927 eclipse [24]. This 
was to use the beads quantitatively, by comparing their times of disappearance, and 
of appearance, with those calculated from the appropriate Hayn profile. The times 
when a thin thread of light was first (or last) broken by a peak were also used. The 
probable errors were i 0"04 and i 0"06; the method was also employed in 1936, 
1945, 1947, and 1948 [25] but results do not seem to be available. 

(d) Banachiewicz's method still used only the bottoms of valleys and the tops of 
mountains, though it used a considerable number of them and allowed for all their 
different depths and heights; an essentially continuous use of the limb-contour 
can be made in the "spectroscopic" method proposed by Lindblad, and used by 
expeditions directed both by him and by Lttndmark in 1945. In this, slitless spectro- 
grams were taken as rapidly as possible of the thin crescent near second and third 
contacts, and to measure them Kristenson [26], following Lindblad, ran a photo- 
meter across the spectrogram at a region where there were no appreciable chromo- 
spheric lines. The change of wavelength (and so of intensity) involved in travelling a 
fairly small fraction of one solar radius in the direction of the dispersion is small, and 
can in any case be determined ; accordingly, running the photometer in a straight line 
across the spectrum (which is convenient and safe) amounts to the same as running 
it round in an arc of a circle parallel to the image of a spectral line. The method does 
not eliminate errors due to bad seeing (as has sometimes been claimed for it) ; but 
the fact that the continuum is spread out by dispersion does allow a sort of rectifica- 
tion of the crescent, as well as neutralizing "photographic spreading" of the image; 
the photometer readings (calibrated, of course, to reduce them to intensities) thus 
give essentially the total intensity as a function of position along the crescent. If 
it were not for solar limb-darkening, the intensity would be simply proportional to 
the thickness of the crescent (reckoned parallel to the dispersion) and should show 
variations directly dictated by the lunar limb-detail; in fact, by comparing the 
intensity at the same limb-point at different times, the limb-darkening itself was 
also determined, in good agreement with Wildt's theory [27], so that the results did 
give a series of true pictures of the actual distance of the lunar limb from the solar 
one, point by point. The probable error of the Moon's place is given as ± 0"023, 



IgO An introduction to the Et-lipw Moon 

± 0'09. This method, as well a* Banacuikwick's, should apparently help to deter- 
mine the extent to which the east ami west limbs of the Moon can hi* fitted to a 
single circle. 

(e) Another method, which has not this particular advantage, but which seems to 
leave very little, room for ambiguity in its interpretation, and which also uses a 
'continuous" run of lunar limb-points, was devised by the writer [28], and tried out 
at Mombasa in 1 948, and, on a more extensive scale, in 1 962. This consists in making 
a timed cinema record, at- a station a little outside I be track of totality, of the rapid 
swing in position-angle shown by the crescent near mid-eclipse. The swing may be 
made so rapid that- a wide range of angles is obtained before the image has moved out 
of the field of a stationary camera, so that the diurnal motion can give the zero of 
position angles very precisely (Fig. I). Owing to the sharpness of the cusps, the 




04: 28: SO 04 1 26:54 04:27:28 04:28:02 04:28:80 04:29:10 

(Greenwich Menu Time) 

Wig. I. six of the 8100 pictenea obtained tit tin- Mombasa eclipse <ti' I si November, 1048. The 

times shown arc onlv approximate, and m*timl tiinfc* are good to a hundredth of *" second or 
better, Tar nil 8100 exposures. The camera was stationary, the Sun wm rising almost vertically, 

and the Moon was rising le*B faxt and viny south. The minimum separation of centres of 

Sun and Moon was just over I'. which tuiminta far tbs rapid rotation 

fluctuations in the run of position -angles, due to lunar irregularities, are marked; 
but the Mombasa fluctuations follow in considerable detail the fluctuations inferred 
from two appropriate limb-traces which were kindly supplied by Watts. The close- 
ness of the fit in detail is sensitive to the value assumed for the Moon's diameter, and 
may even provide a check on its value. The systematic revision which Watts is 
still undertaking has resulted already in a marked improvement in the systematic 
fit, as against that obtained with his first analysis, and it seems probable that the 
method ran locate "Waits' Moon" more accurately, in both co-ordinates, than he 
himself can yet be sure of the true heights of that Moon's mean spherical surface, 
relative to its observed contour. The possible geodetic accuracy is thus greater 
(as may also be true with Lixi>ui,ai>'s method), than the accuracy with which the 
Moon's place can be derived from the results, essentially because one cannot, yet 
sufficiently specify the Moon itself. A detailed analysis of this work is now nearing 
completion. 

In discussing the "probable error" of the place of the Moon obtained by "con- 
tinuous" observations along its contour, one is brought up against a difficulty which 
is also inherent {though less evidently so} in the other methods. The Gaussian 



R. d'E. Atkinson 187 

Theory of Errors assumes that the errors of successive measurements of a quantity 
are independent ; but if one is trying to determine the best mean circle for a given 
lunar profile, and one makes a very large number of radius-measurements, so that a 
hundred of them (say) all fall within the compass of a single mountain, the departures 
from the mean radius are clearly not independent at all; a hundred consecutive 
residuals are all positive. For determining the minutiae of a mountain's shape, a 
hundred measurements within a short length of limb are, of course, better than ten ; 
but as far as the best mean radius for the whole profile goes, they are very little 
better, and they certainly have not got ten times the weight of ten. If we determine 
the co-ordinates of the centre from (say) twenty points equally spaced along the whole 
arc (Hayn, loc. cit.), the twenty residuals may reasonably be considered independent 
(though their distribution will still not be Gaussian) ; but since we know they will 
run up to Jz 2" or more, the "probable error" of our result must be expected to be 
about i 0"15, in good agreement with the values actually claimed. One cannot 
fairly claim to get a much smaller value merely by making many more measurements, 
because (to put it roughly) a change in libration could so easily abolish one mountain, 
or push another up, over a range which would then be long enough to have a systematic 
effect on one's measurements. In the author's Mombasa work, 3100 timed position- 
angles were obtained ; but it would quite clearly be wrong to take the mean scatter of 
the measured position-angles from a smooth run, and to divide it by V3100 and infer 
a probable error of the Moon's place from the figure so obtained ; if one did, one would 
quote i 0"004 or thereabouts, but a slight change in libration would give a result 
different by many times this figure. 

One can, however, legitimately claim a smaller probable error than the 0"15 
quoted above if one is comparing one's observations with those to be expected for a 
Moon of known profile (Kordylewsky, Kristenson, Atkinson). For (assuming 
that the profile is in fact suitable) the differences between the computed and observed 
values can now with much more freedom be called random: measurements at 
successive points along the limb can quite evidently be made much closer together 
than before without the residuals becoming strongly correlated with their neighbours. 
Moreover, the mean residuals will now themselves be appreciably smaller also, which 
would in itself reduce the probable error. It seems difficult to formulate any rigorous 
criterion for the maximum number of observations which it would be fair to regard as 
independent, and certainly any rigorous criterion could only depend on an appro- 
priate theory of (dependent) errors [29]. But bearing in mind how often the Gaussian 
theory is formally applied to cases in which its fundamental assumptions certainly 
do not hold, one might perhaps formally apply it here also, subject to a rule that the 
number of "independent" observations can be taken as no more than twice the 
number of times that the residuals change sign ; this rule evidently holds whenever 
the residuals are random (provided their median is zero), since for each residual there 
is then a 50 per cent chance that the next one will be of opposite sign. The "probable 
error" of the place of "Watts' Moon" (say), as located by "Atkinson's method" 
(say) would then follow from the mean residual divided by the square root of twice 
the number of its changes of sign, in much the same way as probable errors are 
generally derived from mean residuals and numbers of residuals themselves ; neither 
the coefficient 0-8453 (if the mean residual is taken) nor 0-6745 (if the root mean 
square) would be strictly applicable; but that has frequently not prevented their 
use in the past. 



188 An introduction to the Eclipse Moon 

(8) Conclusion 

This paper has stressed the considerable chances which exist at present for systema- 
tic differences between the places of the different Moons. But the application of a 
really uniform system of limb-corrections, such as may be hoped for from Watts' 
work, should greatly help to fuse some of these Moons into one ; indeed, it seems 
probable that a most significant observational advance would be made if the Limb- 
contour Moon and the Astrometric Moon were at once fused observationally (as is 
clearly possible in principle), and were jointly observed as widely as possible from 
now on. The Occultation Moon, if reduced to Watts' sphere, should clearly fall into 
agreement with the Astrometric so reduced, and a rigorous revision of the star-places 
would then allow a reliable comparison with the Eclipse Moon; meanwhile, this 
last one must remain definitely in need of continual further observation in its own 
right. It falls in a quarter of the month that is otherwise quite unobserved; if it can 
nevertheless be relied on to be consistent with the others, it provides a check on the 
Sun's place (and diameter), free from the risk of falsification by refraction. It can 
provide geodetic information [30] ; however, this is true also of the Occultation Moon 
[31], and probably of the Astrometric one in addition. Indeed, the Astrometric 
Moon may well supplant both the others before long in this field; it is not too much 
to say that if Markowitz's method works out as it appears likely to, then as far as 
geodetic work is concerned we shall in future be as well off as if we could order our 
eclipses by the dozen, whenever and wherever we wished. 



Notes and References 



[1] I.A.U. Draft Reports, 1952, pp. 94-98; I. V. Belkovich, Engelhardt Obs. Publ., 24, 242, 1949; 
A. A. Yakovkin, Kiev Obs. Publ., 3, 17, 1950 and 4, 71, 1950. However, see also K. Koziel, Acta 
Astron. (a), 4, 61, etc., 1948-9. 
[2] H. Spencer Jones, M.N., 99, 541, 1939. 

[3] K. F. Sundman, VActivite de la Comm. Geod. Bait., 1944-7, 63, 1948. 
[4] F. Hayn, Abh. d.k. sachs. Oes. d. Wiss. Leipzig, 30 and 33, 1907 and 1914. (Selenographische 

Koordinaten, III and IV.) C. B. Watts and A. N. Adams, A.J., 55, 81, 1950. 
[5] According to T. Banachiewicz, the Saros is the smallest period for an approximate repetition of 

both co-ordinates together. Acta Astron. (c), 4, 160, 1951. 
[6] F. J. M. Stratton, Mem. Roy. Astron. Soc, 59, 257, 1909, and references there. Also [1] above. 
[7] R. T. Cxjllen, M.N., 86, 344, 1926. 
[8] F. M. McBain, Greenw. Obs., 1939 Appendix (re-discusses 1943-7); Astron. J., 53, 163, 1948 and 

55, 7, 47, 247, 1949-51; D. Brotjwer, Astron. J., 51, 144, 1945, etc. 
[9] C. B. Watts, Astron. J., 48, 170, 1940. 
[10] For example, H. Spencer Jones, M.N., 85, 11, 1925. For a general discussion, with other important 

references, see H. Illigner, Astron. Nachr., 253, 293, 1934. 
[11] For the Pleiades, F. Kustner, Nova Acta, 41, 1, 1879; J. Peters, Astron. Nachr., 138, 113, 1895; 

H. Illigner, loc. cit.; for lunar eclipses, L. Struve, Astron. Nachr., 135, 169, 1894. 
[12] M. Wichmann, Astron. Nachr., 33, 309, 1852. 
[13] H. Kobold, Strasbourg Ann., II, Annexe A, 1-39, 1899. 
[14] Ibid., Ill, Annexe A, 1-36, 1901. 
[15] J. Merlin, C.R. Acad. Sci. (Paris), 144, 20, 1907. 
[16] Ibid., 148, 146 (and 263), 1909. 
[17] H. W. Naumann, Astron. Nachr., 193, 123, 1913. 
[18] Ibid., 201, 217, 1915, and 233, 189, 1928. 
[19] F. Hayn, Astron. Nachr., 193, 117, 1913. 
[20] Ibid., 201, 185, 1915. 
[21] Ibid., 233, 183, 1928. 

[22] Some others were: C. W. Wirtz, Astron. Nachr., 171, 102, 1906; E. Becker, Astron. Nachr., 180, 
387, 19; A. Wilkens, Astron. Nachr., 193, 139, 1913; G. N. Neujmin, Mitt. Nik. Obs. Pulkowo, 5, 
77, 1913; M. Simonin, Ann. Obs. Paris, 1914; H. Batterman, Astron. Nachr., 202, 23, 1915; 
C. le Morvan, Bull. Astron. (Paris), 32, 337, 1915; Moscow Obs., Russian Astron. J., 2 (2), 70-76, 
1925; (various) Bull. Astron. Inst. Netherlands, 3, 266, 1927; S. Ishii, Tokyo Astron. Bull., 5-6, 
1927; (various) Bull. Astron. Inst. Netherlands, 4, 71, 1927; E. Przybyllok, Astron. Nachr., 
231, 293, 1928; E. Perepelkin, Astron. Nachr., 232, 127, 1928; S. Ishii, Japan J. Astron. Geophys., 



Harold Jeffreys 189 

11, 37, 1933; H. G. Lengauer, Proc. of Eclipse Expeditions 1936, pp. 129-144, published by 
Russian Academy of Science (in Russian); J. E. Willis, Astron. J., 47, 109, 1939; S. Ishii, Tokyo 
Astron. -BwW.,426-7, 851, 1940; G. G. Lengauer, Reports of Moscow- Leningrad Eclipse Expeditions, 
vol. II, 137, 1941. 

[23] F. Vles and J. Carvallo, C.R. Acad. Sci. (Paris), 154, 1142, 1912, and 155, 545, 1912; Da Costa 
Lobo, C.R. Acad. Sci. (Paris), 154, 1396, 1912; A. de la Batjme-Piajvinel, ibid., 1139; E. Car- 
vallo, ibid., 1072; R. Schorr, Astron. Nachr., 191, 429, 1912. 

[24] K. Kobdylewsky, Acta Astron. (b), 1, 133, 1932. 

[25] T. Banachiewicz, Acta Astron. (c), 3, 16, 1936; I. Bonsdorff, VActivite de la Gomm. Geod. Bait., 
1943, 12(1944); B. Lindblad, ibid., 17; H. Kristenson, Stockholm Obs. Ann., 17, pp. 15-16, 1951; 
I. Bonsdorff et al., Proc. Finn. Acad. 1948, 99 (1950); W. Kinney, Nat. Oeogr. Mag., 95, 325, 
1949; T. N. Panay, Trans. Amer. Geophys. Union, 31, 809, 1950. 

[26] H. Kristenson, Stockholm Obs. Ann., 17, 3, 1951. 

[27] R. Wildt, Ap. J., 105, 36, 1947. 

[28] R. d'E. Atkinson, M.N., 113, 18, 1953, and later papers. 

[29] H. Jeffreys, Theory of Probability, Cambridge University Press, pp. 229-245, 1939. 

[30] I. Bonsdorff, VActivite de la Comm. Geod. Bait. 1942-3, 12 (1944). 
31] J. A. O'Keefe, Astron. J., 55, 177, 1950. 



The Moon's Principal Librations in Rectangular Co-ordinates 

Sin Harold Jeffreys 

St. John's College, Cambridge 

Summary 

The Moon's librations of the first and second orders are discussed by a method that involves direct use of 
rectangular co-ordinates and Lagrange's equations of motion. The secular stability in presence of 
dissipation of the type that might arise from internal tidal friction is also discussed. 



All the textbooks on celestial mechanics that I know deal with the Moon's librations 
by expressing its angular position in terms of Ettler's angles, using Ettler's dynamical 
equations, and then making various more or less cumbrous transformations to reduce 
the problem to one of small oscillations. As the direct use of rectangular co-ordinates 
seems to be gaining popularity in planetary theory, and has already gained it in 
lunar theory, I think that it may be of interest to show how the chief results for the 
librations can be found by using rectangular co-ordinates and Lagrange's equations 
from the start. 

The main difficulty comes from the fact that at least one second-order libration is 
on the verge of being observable, and consequently it is necessary to have the kinetic 
energy T and the work function W to the third order of small quantities. Since the 
principal rotation is not small, it needs to be known to the third order ; the other 
angular velocities are needed to the second order. 

The quantities observed most directly are the displacements with respect to the 
limb of a point near the centre of the disk. A third co-ordinate is provided by the 
change of position angle of a known mark near the limb with respect to the centre. 
If we take the principal axes at the centre of the Moon, OA pointing nearly to the 
Earth, and OC the principal axis nearly normal to the orbit, we can refer them to a 



Y 


Z 


sin </> cos 6 cos ip 


— sin 6 cos y> 


+ cos (/> sin ip 




— sin (f) cos Q sin ^ 


sin 6 sin y> 


+ cos <f> cos y> 




sin cf> sin 


cos 



190 The Moon's principal librations in rectangular co-ordinates 

set OX, OY, OZ, where OZ is fixed in direction and OX, OY rotate about it with 
angular velocity n, OX pointing nearly to the Earth. The usual practice is to choose 
Euler's angles so as to specify the position of C first and then introduce a further 
turn about C ; thus the displacements of A in latitude and longitude, which are 
directly measurable quantities, are each the sum of two parts. It would be more 
natural to specify the position of A first and then B. We need the matrix of direction 
cosines of OA, OB, OC with respect to OX, OY, OZ, expressed in terms of the small 
quantities of the problem, and some at least of the elements are needed to the third 
order. For the usual Euler angles 6, <f>, ip this matrix is exactly, (Jeffreys, 1950) : 

X 

B = A' /cos <f> cos 6 cos ip 

I — sin (f> sin ip 

C = B' \ — cos <j> cos 6 sin ip — sin </> cos d sin ip sin d sin ip j .... (1] 

\ — sin cf> cos ip 

A = C \ cos ^ sin 6 

A', B', C are the axes as given in the book. Take sin <f> = %, Q = \tt + sin -1 g, 
ip = I77 -j- sin -1 r], expand to the third order in %, £, r\, and rearrange. We have 



(2) 



It is simple to start with the elements of zero and first order and apply corrections 
in succession to make the matrix orthogonal to the third order ; but there are some 
ambiguities, which are most conveniently resolved by choosing the higher terms to 
that the velocity components do not contain x explicitly (this being the chief good 
point of the Euler co-ordinate <j>). This object is achieved automatically by direct 
transformation of the exact matrix. 

The space velocities of C with respect to X, Y, Z are to the second order 

£ + XV + XV — n (— V + *£), — V + %! + X% + w(£ + %r\), — ii — r\r\, 
and resolving along OA, OB we have two components of angular velocity 

^2 = i + ( n + X)n> —o> 1 = —rj + (n-\- %)£. (3) 

The velocity components of A are 

-xz-ei- n{x - hn X - ¥H ~ X& + n(l - \x 2 ~ W\ ~ t 
and resolving along OB gives, to the third order, 

*>3 = n + x + \x 2 x - W> + m 2 + v 2 ) -nl • • • .( 4 ) 





X 


Y 


z 


A 


I 1 - k 2 - \? 


X zX£ 


-i 


B 


- x + tn + Wx 


1 - k 2 - W + &nx 


r\ — \¥r\ 


C 


U + ct-W + n 2 ) 


— r} + x% + \vx 2 


1 - \? - w 



Harold Jeffreys 191 

Then 2T = Ay) 2 + B£ 2 + Cf - 2nA£rj - 2n(C — B)£rj 

- (n 2 + 2ni){(C - A)£ 2 + (C - Ufa 8 } 

- 2A^i - 2{C - B)4%. ....(5) 

Constants and derivatives with regard to t have been dropped. 

The direction cosines of the Earth with respect to OX, Y, OZ are taken to be 

— {1 — \ sin 2 i sin 2 (v + pt)} cos (v — nt), — sin (v — nt), — sin * sin (v + pt), 

where v is the Moon's longitude, — p the motion of the node, and i the inclination. 
v — nt is small of the order of the eccentricity e. It is easily verified that the sum of 
squares differs from 1 by a quantity of order i 2 e 2 . Then we are treating i and e also 
as small quantities of the first order. Then with respect to OA, OB, OC the direction 
cosines are, (V to the third order and m', n' to the second), 

V = — {1 — | sin 2 i sin 2 (v + pt)} cos (v — nt)(\ — \% 2 — -|£ 2 ) 

— i sin (v — nt) -\- | sin i sin (v -\- pt) . . . 

m' = % — £r] — sin (v — nt) — r] sin i sin (v -\- pt) 

n' = — £ — yrv] + r) sin (v — nf) — sin i sin (v + pt) 

V can be eliminated at once from the work function. If R is the actual distance from 
the Earth and a the mean distance, the relevant part of the work function is 

W = (- J {A + B + C - S(Al' 2 + £m' 2 + CV 2 )} 



n e 



(^\ 3 {B + C - 2A - 3(B - A)m' 2 - 3(0 - A)n' 2 }. 



2(1 + p) 

Dropping terms independent of £, r\, £ we have 
3n 2 



(7) 



TT= - 



2(1 + ) (1) K 5 ~ ^& _ Sin (V _ ^ — ^ + Sin * Sin ^ + ^^ 
+ (C — A){£ + sin i sin (v + ^) + r\{% — sin (v — nt))} 2 ] 

3n 2 / a\ s 
= - ^) [(^ - il)fe - sin (v - ^)} 2 + {C-A){§ + sin * sin (t; + pt)} 2 

+ 2(0 — B){£ + sin i sin (v + ^)}{;t — sin (v — ^)>y] (8) 

Forming Lagrange's equations to the first order, and putting 

C - A = Bfl, - B = Aol, B - A = Cy .... (9) 

with a + y — — a£y = 0, (10) 

we have the equations 

'I - n{\ - P)rj + (4 - t-^-) n 2 # = - -~- sin i sin (v + pt) . . . .(11) 

t] + n{\ — a)| + w 2 a^ =0 (12) 

it+^Z = i^ sin(B _ M ,) .... (13) 

1 + // 1 + /I 



192 The Moon's principal librations in rectangular co-ordinates 

To this order v -\- nt = (n -j- p)t, v — nt = 2e sin (n — g)t, where g is the motion of 
perigee. Since 1 ^> pjn ^> a, /?, y, the first approximation is, as usual, 

67i 2 ye 

£ = P sin (n + p)t, rj = — Q cos (n + p)t, % = r^—. r, ^ sin (n — gr)J, 

(1 + p,)(n — g) 2 

....(14) 

where 

3 ^ sin * ., g = ^ sin * - (is) 



(1 + ^)(2p/n - m " (1 + ^)(1 + p/n)(2p/n - 3/5) 

On account of the small divisor 2p/n — 3/3, P and Q are of the same order of magni- 
tude as i. x nas no small divisor (we are not at present considering the annual term 
in v — nt). The speeds of the free motions are approximately n(l -\- •§/?), 2n\/(u.(i), 
and n<s/(3y). 

In proceeding to a second approximation we use 

v — nt = 2e sin (n — g)t, (a/E) 3 = 1 -f- 3e cos (n — g)t, .... (16) 

sin (v + pt) = sin (n + p)t — e sin (p -\- g)t + e sin (2n -\- p — g)t ... . (17) 

Second-order terms may rise in importance if their speeds are small or near n. In 
T, the term — A£r)% does not contain the small factors a, /?, y. Substituting the 
first approximations to £ and 77 we find that it contributes to the left of (13) a term 
— n(n -\- p)P 2 sin 2(n + p)t. The contribution of the terms on the right is 

3tj2 

— — AxQ(P + i) sin 2(ti + p)t, .... (18) 

1 + // 

which is much smaller on account of the factor a. Then the second-order part of % is 

\P 2 sin 2(to + p)t. (19) 

In the third-order terms in W, x is much less than v — nt, and may be dropped in 
forming the equations for £ and r\. Terms in e in sin (v + pt) must be retained. 
The extra terms in W are 

372/^6 

[3P/?{£ + sin i sin (n + #)£} 2 cos (ti — g)t 



2(1 + ^) 

+ 2B(3{£ -f- sin * sin (ti + p)£}{sin (2ti -f- p — g)t — sin (p -j- £7)0 sin i 

— 4 A a{£ + sin « sin (w -)- p)t}rj sin (w — </)£]. .... (20) 



mi • ^ W 

lhe new terms in -r— are 



3 71/ 6 

(ffijSP + iJ3# - ^aQ) sin (p + g)t .... (21) 



and in ~^— 

CYj 



1 +/* 



3ti 2 6 

,4<x(P + sin i) cos (p + g)t. (22) 



1+^ 



Terms of speed 2n + p — g have been dropped because, being fortnightly, they are 
not subject to magnification. 



Harold Jeffreys 193 

The term — A£r)x in T still needs consideration. It adds to the left of the £ 

equation 

3n 2 
Ml * t~. — AQye sin (p + g)t .... (23) 

1 + fi 

and converts the Act of (22) into Aft, which we can replace by Bft* 

The last change is comparable with the effects of terms in C — B and B — A 
on the right, which we have not considered. The effect on the r\ equation contains 
the derivative of the slowly varying factor, and is negligible. If we also, as usual, 
neglect the difference between P and Q the additional terms on the right of (11) 
and (12) reduce to 

3 Yl^&B 3%^60C 

— o ^—, — ( p + sin i) sin (p + g)t, — — (P + sin i) cos (p + g)t (24) 

The largest terms on the left are those containing £ and rj, and integration gives the 
Poisson terms 

3nea. P + sin i . 

£ = — ^— — sin (p + g)t, . . . .(25) 

1 + n p + g 

3 neft P + sin i 
y=-f r-^~ rn , n cos (p + g)t. . . . .26 

2 1 + ^ p + g 

i 
I and ^ are usually denned as displacements of C toward X and Y, and therefore 

correspond to the present £ + yv\, — r\ + %£\ but the second-order terms do not 

contain the small divisor p -\- g and are therefore negligible. 

The amplitudes of the Poisson terms in f and rj are of the order of 1-4 ; Tissekand 
gives 947" and 972", apparently through a misplaced decimal point, and the mistake 
is not indicated in the errata. Plummer's Dynamical Astronomy gives the correct 
value. 

The annual term in y, of speed n' (which will not be confused with the direction 
cosine), leads to second-order terms in £ and rj. The most important come from the 
A£tjx term in T, but the differentiation of % introduces the small factor n' , and the 
terms in £ and r\ have speeds n -\- p ^ n' and are less magnified than terms in 
(n + p)t. Hence these terms are negligible. 

The conditions for ordinary stability are y > 0, ocft > 0, so that the Moon would be 
stable either for G > B > A or for B > A > G. If, however, there is small damping 
proportional to the velocities relative to the mean velocity of rotation, such as might 
occur through tidal friction, the equations of free motion become 

£ + nk£ - n{\ - ft)rj + U - -^-) n*ft£ = 0, 

rj + nkrj -\- n(l — <x)£ + n 2 caj = 0, 

, 3w 2 y 

X + nkx + — — = 0. 

1 + jU 



* Euler's equations, of course, give Bfi directly at this point, essentially because they refer to rotations about exactly 
rectangular axes. 



194 A general expression for a Lagrangian bracket 

The time factors in | and r) are, approximately, exp At, with 

n 



That in % gives 



-== -l*±'»V(3y)- 

71 



a/? > for ordinary stability ; but if a, /? were both negative there would therefore 
be secular instability in £ and r\. Hence the Moon could be secularly stable only in the 
actual condition C > B > A. 

The direct use of the matrix (2) reduces much of the work to routine ; actually 
only the direction cosines of A are needed to the third order. Once the angular 
velocities and V , m' , n' are known in terms of $, rj, %, Euler's equations are probably 
slightly simpler to use than Lagrange's. Lagrange's have the advantage that (8) 
is still available if perturbations of the orbit are to be considered. The problem does 
not look hopeful for the application of "vector methods". 



Reference 

Jeffreys, H. and B. S 1950 Methods of Mathematical Physics, Cambridge, 

p. 123. 



A General Expression for a Lagrangian Bracket 

W. M. Smart 

University Observatory, Glasgow 

Summary 

In the paper, based on an earlier investigation by A. Y. G. Campbell, the general expression — in terms 
of any functions of the orbital elements — for any one of the Lagrangian Brackets which are constituents 
of Lagrange's planetary equations, is derived. 



(1) One of Professor Stratton's earliest researches (1909a, 1909b) was in the 
strenuous discipline of celestial mechanics on the subject of The Constants of the 
Moon's Physical Libration ; in joining with so many of his friends and admirers in 
saluting him on his retirement from the Chair of Astrophysics, I think it not 
inappropriate to offer a small contribution in the field of his original interests. 

(2) A Lagrangian Bracket, [u, v], is defined by the expression 

d{x, x) 
[u, v] = 2 Tj r» ••••(!) 



W. M. Smart 195 

where x, y, z are the rectangular co-ordinates of a planet and u, v are usually taken 
to be two of the six elements of the orbit — a, e, i ; Q., a>, and e in the well-established 
notation ; each co-ordinate is a function of the time and the six elements. The 
fifteen brackets (1) are constituents of the six Lagrangian equations of planetary- 
motion and their evaluation, when u and v are elements, is found in the principal 
works on celestial mechanics. 

The present note is based on a method first suggested by Campbell (1897), over 
half a century ago, and is substantially in the form in which I have given it in lectures 
for some time past. We shall assume that, in general, u and v are any independent 
functions of the six elements. 



(3) Let 



„ dx dy dz 

F »=*Tu + y£ +z ^ •• (2) 



with a similar expression for F v ; then, from (1), 



oF u oF v 

[ «- r] = -* - *r- •••• (3) 



Now 



also 



where 



dF\ 
dt 



z,y,z 

x 2 + y 2 + * 2 = fi (- - -) = 27 + 2F , 
\r a J 



V = £, V = - £- and p = n 2 a*. . . . .(4) 

r 2a 



The equations of motion in the elliptic orbit are 



Hence 



dF\ 
dt 



jux .. 3F ± 

x -f- - — = 0, etc., or x = -^—, etc. 
r 3 ox 



2g-3 + s" + '•>-*<" + '•> 



x,y,z 



Integrate between t and t, where t is the time of perihelion passage ; then, 

Then, if 

(V+ V )dt, 



: 



196 A general expression for a Lagrangian bracket 

we have, since t = (e — co), 

n 



s-f^ + ^-^w + ^f;- 

Hence 

'.-*.w = 4 + 2 ^ T)+F «>S- ( '- T) S (5) 

Now, in terms of the eccentric anomaly E, r is given by 

r = a(l — e cos E) ; 
hence, since E = when t = t, V(t) — ju/{a(l — e)}, and (5) becomes 

J. -*» = 2- + j^— -. __((_ T )_ , 
which we write as 

2 £-'i&- J - = c « ••■•< 6 > 

where 

"* ' o(l -e) ^ 3w l } 

Here C M is a constant, being a function of the elements. Similarly, 

where C v is denned in an analogous way to (7). 

Differentiate (6) with respect to v and (8) with respect to u, and subtract ; then, 
from (3), 

i-^-g-S- •■■■<•» 

Since the right-hand side of this equation is constant, the characteristic property 
of a bracket is deduced, namely, that [u, v] does not involve the time explicitly. 
The evaluation of [u, v] requires the evaluation of -^(t) in (7). 

(4) The three angular elements of the planetary orbit with which it is convenient 
to begin are : 

D, = YN, co = NA, and * (the inclination), 

where N is the ascending node and A is the direction of perihelion on the sphere, the 
Sun being at the centre 0. 

Let (f, rj) be the co-ordinates of the planet with respect to axes OA, OB in the 
orbital plane (NB = 90° + co). 

If (l v m x , riy) and (l 2 , m 2 , n 2 ) are the direction-cosines of OA and OB with respect 
to the original axes, then 

x = IJ + l 2 r\, y = m^ + m 2 r\, z = n^ + n 2 rj ... (10) 



W. M. Smart 197 



We have : 



and 



l x = cos Q, cos co — sin Q. sin co cos i, 
m x = sin O cos co + cos Q sin co cos *, 
% = sin co sin *', 

l 2 = — cos O sin co — sin Q cos co cos i, 
m 2 = — sin Q sin co + cos O cos co cos **, 
w 2 = cos co sin i. 

Also, if (l 3 , m 3 , n 3 ) are the direction-cosines of OC, the normal to the orbital plane, 
then 

l 3 = sin Q. sin i, m 3 = — cos Q. sin *, n 3 = cos *'. 



From these 



Hence 



Similarly, 



dl x dm x dn x 

3Q = _miJ aa =?1 ' 3Q = 0; 

3Z X 3m x dn x 

3Z X . 3m, . 3% 

— = l 3 sin <w, — - = m 3 sin co, — - = n 3 sin ft). 
o% dl dl 



dl x dl x 3D. dl ± dco dl ± di 

du dQ. du 3ft) du di ' du 

dQ, dco . di 

= — m 1 — + l 2 — + l 3 sm co — • 

dw dW dw 



3m x 30 3co d* 



3% 3ft> . di 

3~ = w a y- + to 3 sin co — 

dw dU dU 



From these and the relations between direction-cosines, 



l,m,n 3% du du 

Also, since SZ X 2 = 2Z 2 2 = 1, and HZ^ = 0, then 



^ , dl-, 3D 3ft) 



du du 

and 



S,^ = S.||=0 ....(12) 



^sh-^-s- -< 13 > 



198 A general expression for a Lagrangian bracket 

(5) From (2) and (10) we have 

/ dg dh T dri dO 



* + * 5S + «"» - &> »• *■ 



by means of (12) and (13). 
But, 



where ft is twice the rate of description of area in the elliptic orbit ; hence by means 

of (11), 

-, 3co T . dQ. ., .. 

+ Ifi 1- h cos * - — (14) 

du du 



'*>-l's + «s 



[< 

Now, 

| = a (cos E — e),tj = aVU — e2 ) sm E> 



$ = — and ^7 (t) 



also, Kepler's equation is 

E — esin E = n(t — t), 
from which 

Then, 
Also, 
from which 

Again, 



r 1 — e 

a sin 2? . &, so that |(t) = 0. 
i) = a^/(l — e 2 ) cos .6/ . E, 
7Wb\/{\ — e 2 ) 



^(t) = 



1 — e 



S)-^-"©.- 



From Kepler's equation 



r dE . de dn dr 

a du du du du 



hence 



(dE\ __ n dr 

\du) T 1 — e' du 



Inserting these results in (14) we obtain, with ju given by (4), 
Hence (7) becomes 



all + e) dr 7 dco 7 . dQ 

MV a(l — e) dp dw dw 



dF 7 doj . .dQ .,_. 

M du du du 



W. M. Smart 199 

or, in terms of a>, s, and replacing V by , 

*.-£<.-a>£-»£ + »(i— o£ <*> 

Campbell's procedure for evaluating the Lagrangian Brackets when u and v are 
any two of the elements a, e, i, £1, co, and e may be interpolated here. From (16), 

C « = I? (C ~ S) ' C e = C i = > 

C Q = A(l - cos »), C~ = —h,C e = 0. 
Then, for example, 

ac e dc a 

[o, e] = -^ - -^ = 0. 
da de 

In this way the non-zero brackets are found to be, on writing e = sin <f>, 
[a, e] = — \na, [a, Q.] = \na cos </>(l — cos i), [a, eo] — \na{\ — cos (f>), 
[e, Q] = -- na? tan <f>(l — cos i), [e, d>] = ?ia 2 tan 0, [i, £1] = na 2 cos <f> sin i. 

(6) The general expression for a bracket can be derived at once from (9) and (15), 
u and v being any functions of the elements ; it is, with V = — —, 

[,,„]= y ' *; + m> + *°- * T - > , ••••< i7 » 

d(u, V) d(u, V) d{u, V) 

or, in terms of d> and e, 

a ( e ~™ _ Ji\ 

\ n ' 2a) d(a> — Q, h) 3(Q, h cos i) 

[u, v] = 1 1 (18) 

d(u, V) d(u, V) d{U, V) 

In particular, if u and v are any two elements the evaluation of the corresponding 
bracket can be quickly obtained. 

(7) The formula (17) enables us to derive the canonic equations expressed in the 
usual forms. 

Introduce <x r and f} r (r — 1, 2, 3) given by 

a x = — — , a 2 = h = ^[/ua(l — e 2 )}, a 3 = h cos i, 
za 

/? x = — T, /5 2 = CO, /? 3 = Q. 

Then (17) becomes 

[u, v] = — 2 -37 r- 

r ti d{u, V) 



200 The role of cracovians in astronomy 

Hence, since the a's and /?'s are independent, 

[<*,,&]= -1 ....(19) 

and 

K, a J = [fi r , &] = K, J = 0, * # r. .... (20) 

If R is the disturbing function, Lagrange's planetary equations in their general 
form are : 

s * . dR 

1 [«r» a »]*8 + 2 [ a r> &]& = ^— . ( 21 ) 

2 [fi r , a s ]a g + 2 [&, ptf, = H .... (22) 

s=l « = 1 ^Pr 

Then (22) and (21) become, by means of (19) and (20), 

* r ~ a&' ^ r ~~ ~ v 

These are the canonic equations in which a r and /? r are a pair of conjugate variables. 



Refekences 

Campbell, A. Y. G 1897 M.N., 57, 118. 

Steatton, F. J. M. 1909a Mem. Roy. Astron. Soc, 59, Pt. IV. 

1909b M.N., 69, 568. 



The Role of Cracovians in Astronomy 

T. Banachiewicz 

Astronomiczne Obserwatorium, Krakow, Poland 

Summary 

The importance and usefulness of cracovians — matrices obeying special rules of multiplication — is 
made evident in the treatment of astronomical problems. To show this, simple applications are dealt 
with in the fields of spherical astronomy, transformation of co-ordinates, and the theory of least squares. 
The discussion emphasizes the theoretical interest of cracovians, and brings out the simplicity, ease, and 
reliability of cracovian operations. 



1. Introduction 

The rise of cracovians in computational astronomy is to a certain degree connected 
with the decline of logarithms. Of course, logarithms continue to play a conspicuous 
role both in the definition of stellar magnitudes and in theoretical developments. 
As regards computations, however, a revolutionary development has taken place, 
which is mainly due to the use of calculating machines (desk calculators). This 



T. Banachiewicz 201 

happened because the desk calculator serves better than logarithms in the formation 
of products of numbers, especially in expressions such as 

p = aA -bB + cC, ....(1) 

for which the desk calculator does not demand the writing out of the separate aA, 

— bB, cC. This has had the consequence that linear expressions like (1), formerly 
almost tabooed in astronomical writings (and often hidden, as, for example, in 
Gauss' Theoria Motus, in the thick of trigonometrical formulae) have now risen to an 
honourable place. We come here to another factor in this development. It has 
turned out, in fact, that further progress may be achieved by a suitable mathematical 
interpretation of the linear expressions, together with a corresponding theory and 
symbolism. For instance, the formulae of common algebra 

I = a£ — br) x — cl — dm X = ex — fy, 

m = a'i — b'rj y = c'l — d'm Y = e'x — fy, 

which give us the relations between X, Y and £, r\, are not very convenient and 
demand a wearisome attention in their evaluation. They may, however, be essentially 
simplified by the use of collective numbers, such as the matrices invented by 
Hamilton and Cayley about a hundred years ago, or the cracovians introduced 
by the writer since 1916. These two types of collective numbers differ theoretically 
only in the definition of the product, and it is natural that mathematicians in their 
desire to deal with a minimum of mathematical entities, would like a 'priori to drop 
the new numbers, or to treat both types in a common theory. The second alterna- 
tive, however, would lead inevitably to errors in the calculations, as it is essential for 
the computer to have single fixed rules. As regards the choice between matrices 
and cracovians, we may quote here some lines from the recent authoritative Polish 
treatise by Hatjsbrandt (1953): "The use (in the book) of cracovians . . .instead 
of matrices . . . does not appear to demand detailed explanations. Everybody 
engaged in practical mathematics knows that the (cracovian) multiplication by 
parallel lines (columns by columns or rows by rows) is in practical computation more 
convenient than the (matrix) multiplication by perpendicular lines. The apparent 
defect of cracovian multiplication, i.e. the absence of the associative law, (ab)c 

— a(bc), is (really) no defect . . . because for the cracovians it is abc = a(c . xb), 
xb being the transpose of 6 . . . . The solution of different problems by cracovians 
has preceded, sometimes by more than ten years, their solution by matrices. This 
appears to indicate the superiority of cracovians as a tool of research". 

Also a number of astronomers from different countries have given preference to 
computations with cracovians, which in Poland are used exclusively. We quote here 
Villemarque (1936) in China, Eckert and Brouwer (1937) in U.S.A., Arend 
(1941-1950) in Belgium, Herget (1948) in U.S.A., and Samoilowa-Jachontowa 
(1949) in U.S.S.R. Hence only the applications of cracovians will be considered here. 

In 1923 the author of this paper introduced the cracovians in print, and developed 
in the following years their theory (Banachiewicz, 1923a, 1949). Multiplication of 
cracovians is carried out according to the following rule : the element p { , in the ith 
column and the jth row of the product/) = a . b of two cracovians a and b is obtained 
by multiplying the ith. column (denoted by a t ) of the first factor a by the jth column 
(denoted by b,) of the second factor b. Thus 

p t , = a i .b i . (3) 



202 The role of cracovians in astronomy 

On the basis of this definition the relation (2) between X, Y and £, r\, becomes : 

denoting the cracovians by the symbols { } and supposing that the cracovians in 
the right member of (4) are to be multiplied in turn : the first by the second, their 
product by the third, and so on. For the numerical values a — + 4, a' = — 1, 
b = + 5, b' = - 6, c = - 5, c' = + 7, d = — 1, d' = + 2, e = + 1, e' = + 3, 
/= + 4,/' = +2,£=+3,?y= + 2, it follows from (2) or from (4) that X = + 15, 
y = + 5. In employing (4) the problem is arranged according to the second precept 
of Descartes (1636) : the given values of a, a', , . . are inserted into the scheme (4) 
and then the operations are carried out with the desk calculator (Banachiewicz, 
1929a). Although the arithmetical operations are the same whether we use (2) or (4), 
it is easy to convince oneself by trial that the use of (4) saves much mental work. 
In such an arrangement of the work lies to a large extent the importance of the 
cracovians as a tool for reducing the mental strain of the computer. 



2. Spherical Astronomy 

The cracovians lead quickly, rigorously, and probably more directly than any 
other way, to the fundamental formulae of spherical trigonometry for a general 
triangle, as well as to those of spherical polygononometry (Banachiewicz, 1923b). 
The underlying concept is the method of "wandering axes". Take a system of 
perpendicular co-ordinate axes OX, Y, OZ with O in the centre of a sphere. When 
this system rotates about the axis OZ through the angle a, or about the axis OX 
through the angle /?, the co-ordinates (x, y, z) of any fixed point M on the sphere 
become : 




[10 \ 

where r(ot) = Jsin a cos a 0|, p{fi) = JO cos fi — sin ^ (5) 

sin /? cos /?J 

Consider a closed spherical polygon with vertices I, 2, . . . n, and let the axis OX 
be directed initially towards the vertex i and the axis OF to a point lying on the 
great circle through the vertices i and 2. Let s i denote the length of the side 
(f, * + i) and a t the angle between the sides (i, i + i) and (i + i, i + 2). Let the 
co-ordinate axes rotate in turn about the axes OZ and OX, first through the angle s 1 
about OZ, then through a x about the new position of OX, and afterwards through the 
angle s 2 about the new position of OZ, and so on. After all these rotations round the 
whole polygon the axes OXYZ will return to their initial positions, and the initial 
and the final co-ordinates of all points M will coincide. 

If we take as M in turn the points (1,0, 0), (0, 1, 0), and (0, 0, 1), we obtain from 
(5) the fundamental formula of spherical polygonometry : 

t . r(s x ) . p[a x ) . r{s 2 ) . . . r(s n )p(a n ) = x, .... (6) 




T. Banachiewicz 203 

where x denotes the unit cracovian 

(1 0) 

....(7) 

In the particular case of a spherical triangle the formula (6) leads directly to the 
familiar eight formulae of Gauss and to the formula of Cagnoli. 

Moreover, the formula (6) is valid mutatis mutandis also if the cracovians r and p 
are replaced by certain other cracovians R and P of the fourth order (i.e. cracovians 
having 4 columns and 4 rows, and depending on the arguments |s 4 - and \a t (Bana- 
chiewicz, 1927). This result furnishes two methods of solving polygonometric 
problems. It extends to spherical polygons the four well-known formulae of 
Delambre of the form : sin \(A — B) . sin |c = sin \{a — b) . cos \C. 

Koebcke (1937), with reference to the above equation (6), wrote as follows: 
"The solution of general spherical triangles was once performed by their decom- 
position into right-angled triangles; only considerably later was the solution of 
general triangles carried out directly. Nevertheless, spherical polygons have pre- 
viously been solved by resolving them into triangles. In this respect the cracovians 
. . . bring about an essential change, and in this domain lies their main theoretical 
significance". 

Formula (6) is frequently applied to calculations concerning the Moon, particularly 
to the determination of the selenographic co-ordinates P and D for occultations or 
solar eclipses (Banachiewicz, 1930). It formed the basis of the differential formulae 
of spherical polygonometry deduced by Koziel (1949). 

Detailed expressions for spherical polygons of 12 or less elements have been given 
by the writer (Banachiewicz, 1948). 

3. Theoretical Astronomy 

The rotational cracovians p and r figuring in (6) are especially useful in practice in 
connection with the transformation of co-ordinates. The vectorial constants of an 
orbit, e.g. are given by the formula (Banachiewicz, 1929b): 

cos SI sin SI 0U1 \ 

sin SI cos SI OHO cose sin ej 

lj (0 — sin e cos ej 

....(8) 

This formula contains 47 symbols (besides the zeros) against 137, i.e. almost 
three times as many symbols in the equivalent common forumlae (see, e.g. Planetary 
Co-ordinates, 1939). The latter are not only less transparent, but they also do not 
contain in themselves the handy computing scheme for formula (8); in contrast 
to the latter they also cannot be written down directly. The transcription of the 
formula (8) using the cracovians R and P was given by Banachiewicz (1928). 

More important are the formulae for computing the differential coefficients of 
geocentric co-ordinates of a planet or comet with respect to the elements of the orbit. 
The notable simplification of the problem brought about by the cracovians is here 
very attractive; see the papers by Kepinski (1927), Zagar (1928), Orkisz (1931), 
Szeligowski and Koebcke (1934), Przybylski (1939), Eckert and Brouwer 




204 The role of cracovians in astronomy 

(1937), p. 132), Moskova (1949), and others. For the case of the elements SI, i, and 
co, the theory was developed by the present writer (1929c) and for Oppolzer's 
elements by American astronomers (see Eckert and Brouwer, 1937, p. 132). 

For the application of the cracovians to the theory of special perturbations see 
Koebcke (1937, p. 13). Different applications to the theory of orbits are to be found 
in the papers of Steins (1950) and the writer (in connection with the problem of the 
accuracy of an orbit; Banachiewicz, 1950) and in many other places. 



4. Least Squares 

1954 = 1 . 10 3 + 9 . 10 2 -f 5 . 10 1 + 4. This universally adopted positional system 
of writing the numbers considerably facilitates the four arithmetical operations on 
them. In all operations on numbers we do not need to trouble about the different 
powers of 10; we forget them. A similar mental simplification may be achieved in 
the solution of linear equations. Since astronomers are interested mainly in the 
solution of the normal equations of the theory of Least Squares, such equations only 
will be considered here. 

Take a system of two normal equations for two unknowns 

4x + 2y = 10. 

2x + lOy =14. (9) 

By introducing the cracovians 



A = \\ io) and L 



-13 

we present the solution of (9) in the form 

ix] =L:A. ....(10) 



y 

The problem is thus reduced to the division of two cracovians. In such a solution 
we have to deal only with the elements of L and A, but not with the equations and 
the unknowns, thus attaining the simplification indicated. 

The division in (10) is made by a resolution of A into a product of two triangular 
cracovians consisting of the upper right-hand half of the square array (Banachiewicz, 
1938), i.e. with zeros below the leading diagonal. This operation, made solely on the 
basis of the single equation G t . //, = A i5 , is very easily performed with a desk calcu- 
lator. One supposes usually G = H — VA ("method of the square root"). One can 
also take G as having units on its leading diagonal ("method of proportional factors") ; 
the method of Doolittle then leads, less conveniently, to the same numerical 
operations. 

The numbers p x and/> 2 > satisfying \ pl \ VA = L, are also found. In our example, 
where VA = j J, p 1 = 5, p 2 = 3, it follows that 

2x + y — 5, 3y = 3, and hence x — 2, y == 1. ... .(11) 



T. Banachiewicz 205 

This is the determinate solution, which was never a matter of any great difficulty. 
It was presented by Cholesky (see Benoit, 1924) in ordinary algebraic language, and 
therefore without the advantages of the operations with collective numbers. 

Denoting by q the inverse of r = V A, so that q = r _1 , one obtains the indeter- 
minate solution of (10), i.e. the cracovian Q in the equation | \ = L . Q, namely 

Q = q 2 ....(12) 

in our example 

y "l-i 1/ - 18 l-l +2 

Gauss was much occupied with a problem which was equivalent to the determina- 
tion of Q, but his results have proved in practice to be somewhat difficult to apply. 
In spite of some misstatements an equation equivalent to (12) was never obtained by 
Cholesky. We note that a still shorter method of calculating Q is based on the 
equation Q = q : r, demanding a knowledge of the diagonal elements of q only 
(Banachiewicz, 1939). 

Besides facilitating computation, the cracovians also simplify the theory of least 
squares. The expression (10), for instance, leads at once to the formulae 



J = L . xq . q = (L : xr) . q = (L : xr) : r, 



each of which would demand for its derivation a lengthy procedure using ordinary 
algebra. The numerical determination of the unknowns may be carried out in a 
variety of ways. 

"In Poland the Gauss algorism has been almost wholly replaced by the cracovian 
calculus", writes Lesniok (1953). 



5. "The Ceacovian Idea" 

The simplification of the solutions of linear problems and the reduction of the 
mental work involved in computations, as achieved by the cracovian calculus, has 
encouraged some workers to introduce other tabular numbers or symbols (relating 
to tables) for the solution of non-linear computational problems. Kochmanski 
(1952) established his "nuclear algebra", dealing with operations on power series in 
two or more variables. Koziel (1954), using the concept of the "nuclei", solves 
with respect to x and y the equations 

£ = ILA^xhf, r\ = Y.B^xY, 

where A tj and B {j denote the known coefficients. Hausbeandt (1953, p. 67) invents 
special auxiliary symbols for geodetic computations. 

Everybody who familiarizes himself with cracovians, and proceeds to practise 
them, will become their friend, often further developing their basic ideas. 



References 

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del et Terre, 57, No. 12, pp. 497-515. 



206 



The r61e of cracovians in astronomy 



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Steins, K 1950 

Szeligowski, S. and Koebcke, F. ... 1934 

VlLLEMARQUE, E 1936 

Zagar, F 1928 



"Etablissement par voie raccourcie des formules 
de Thiele-Innes, relatives aux orbites d'etoiles 
doubles, en recourant aux principes de 
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"Determination of the Position of an Orbit", 
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Acta Astronomica (Krakdw), Ser. c, 1, 64. 

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"Coordonnees selenographiques relatives aux 
occupations", Acta Astronomica (Krakdw), 
Ser. c, 1, 127-37. 

Bull. Acad. Sci. (Poland), A, pp. 393-404. 

"Computation of Inverse Arrays", Acta Astro- 
nomica (Krakdw), Ser. c, 4, 30. 

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et polygonos spherico", Rocz. Astr. Obs. Krak. 
(SAC), 19. 

"Les cracoviens et quelques-unes de leurs 
applications en geodesie, Cracow Obs. Reprint, 
25. 

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Bull. geod. internal., No. 2. 

Discours de la methode; Deuxieme partie. Paris. 

Astron. J., 46, No. 13. 

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Warszawa, pp. 6, 53. 

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Astronomical Applications of the Cracovians). 
Poznan, p. 1. 

Bull. Acad. Sci. (Poland), A, pp. 1-16. 

Acta Astronomica (Krakdw), Ser. a, 5 (in press). 

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Acad. Sci. (Poland). 

(For the years 1940-1960, referred to the equinox 
of 1950-0), 150 pp., H.M. Nautical Almanac 
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Astronomica (Krakdw), Ser. a, 4, 34-60. 

Bull. Inst. Astron. Acad. Sci. U.S.S.R., 4, No. 6. 

Acta Astronomica (Krakdw), Ser. a, 5, 19-36. 

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Mem. R. Istituto Veneto d. Sci., 29, No. 8, p. 40. 



Regularization of the Three-Body Problem 

G. Lemaitre 

Universite de Louvain, Belgium 



1. Introduction 

The motion of three point-masses under their mutual attraction, according to 
Newton's law, is one of the oldest and most celebrated problems of celestial 
mechanics, and is known as the "Three-Body Problem". When two of the bodies 
approach one another, the attractive force becomes very large and for an actual 
encounter it is infinite. In that case the equations of motion lose their meaning : 
encounters are singularities of the problem. In the case of binary collision, that is to 
say when only two bodies collide, it has been possible to define a modified problem 
which has no singularity for binary collisions and is completely equivalent to the 
former problem where this was regular. Such a substitution of a regular problem for 
the singular one is called a regularization of the problem. 

An older example of such a regularization is afforded by the problem in which 
one body only moves under the Newtonian law due to the attraction of the two other 
bodies, which are constrained to remain fixed. This problem has been solved by 
Euler, and Jacobi refers to it as the masterpiece of this great mathematician. It is 
better known in the form in which Lagrange has presented the solution. This is 
essentially a change of co-ordinates, which introduces the so-called elliptical co- 
ordinates. It is a conformal transformation of the network of rectangular co-ordinates 
of which each centre is a singular point. Its effect is that when a trajectory of the 
original problem makes a sharp bend passing near to a centre in a parabolic orbit, 
its transformed image follows nearly a straight line. 

This would not be sufficient to regularize the problem because this straight fine 
would be described with infinite velocity. To complete the regularization it is 
necessary to transform the time by introducing a kind of spurious regraduation of it, 
so that from the point of view of this regraduation the motion would be slowed down 
when approaching the singularity, the new speed remaining finite. The new variable 
introduced in this way is called a regularizing time. 

This process of Lagrange has been extended by Thiele to some particular cases 
of the Three Body Problem and it has been extensively used in practical computation 
in the work of the Copenhagen school under E. Stromgren. A similar process has 
been introduced by Levi-Civita and applied to the motion of three bodies in a 
plane. It is the purpose of this paper to extend such a regularization process to the 
general case. 

Sundman has shown that from a pure-mathematical point of view, the regu- 
larization of time was the essential part of the process, the rectification of the 
trajectory being a mere luxury. But nobody has been able to apply Sundman 's 
theory to an actual numerical problem. It is possible that it is the introduction of the 
superfluous conformal transformation which makes the whole difference between a 
mere mathematical theorem and a powerful numerical tool. 

207 



208 Regularization of the Three -Body Problem 

2. Geometrical Aspects of the Regularization 

The nature of the regularization of the binary collisions in the Three-Body Problem 
can be conveniently exhibited in the simpler case of the Two-Body Problem of 
Keplerian motion. The Hamiltonian for that case, 

// = 1,™ 2 l 1^2 ^_ 

is transformed by a conformal transformation, written 

in complex-variable notation, or 

x = | 2 — r\ 2 , 

y = 2 %y 

in real Cartesian co-ordinates. 
Then 

dx 2 + dy 2 = 4(£ 2 + if){dp -f dr?) 

and therefore the Hamiltonian becomes 

- 2 4(i 2 + t] 2 ) i 2 + n 2 

In order to regularize we introduce a new Hamiltonian 

H ' = (H - A)4(£* + r) 2 ) == UP, 2 + P n % ) - ^ - 4 ^(^ 2 + »? 2 ) = o, 

for which the canonical equations will be valid if they are written with a new time t, 
in place of t, defined by 



Wr 



We have, in fact, 



dr d Pi Ps ' dt dr 2 di 

This is the device introduced by Levi-Civita.* 

It connects the given problem with a new one by a transformation of co-ordinates 
involving the introduction of a new time t. This transformation is regular every- 
where, except at the singularity of the first problem. Therefore regularization means 
replacement of the given problem by another which is regular everywhere and is 
equivalent to the first one except at the point of singularity. 

There are two aspects in this regularization process : first the introduction of a 
regularizing time t and secondly a geometrical aspect of the transformation which 
realizes a simplification of the shape of the trajectories in the neighbourhood of the 
singularity. Mathematically speaking, the new graduation of time by t is the only 
essential point. Sundman's realization of this enabled him to regularize the general 
Three-Body Problem, at least when three-body-collisions are excluded and he 
has shown that this always occurs when the total angular momentum does not 

* See, for example, Happbl (1941), p. 516, and Whittaker (1937), p. 424, where references to the original papers of Levi- 
Civita and Sundman are given. 



G. Lemaitre 209 

vanish. Nevertheless, his process has found no practical use in numerical computa- 
tions. This may be related to the fact that, although Sttndmann accounted for the 
required regraduation of the time he completely neglected the secondary aspect of, 
Levi-Civita's regularization, i.e. the geometrical simplification of the trajectories 
in the neighbourhood of the singularity. 

A confirmation of his point of view may be afforded by the practical interest of the 
process of regularization, introduced by Lagrange in the problem of two fixed 
centres (cf., e.g., Whittaker (1937), p. 91), which has been used with so much 
success by Thiele and the Copenhagen school in the problem of two rotating centres. 

This process makes use of the conformal transformation 

z = cosh £ 

or x = cosh £ cos r\, 

y = sinh £ sin r\. 

Using Whittaker' s notation with c = 1, it leads to the introduction of a new 
Hamiltonian, 

H' = (H — &)(cosh 2 | - cos 2 rj) 

= Pi 2 — (f* + f*') c °sh f — 2h cosh 2 | + p n 2 + (/u — /u') cos rj + 2h cos 2 r\ = 0, 

to be used with the regularizing time 

dt 



/ 



cosh 2 £ — cos 2 r] 



Both collisions with each of the two centres at | = and rj = ± rr/2 are regularized. 

The object of the present paper is to make a regularization of the binary collisions 
which would be efficient for the general case of the Three-Body Problem and would 
include both aspects of the device of Levi-Civita. This has been applied only to the 
problem when the three bodies remain in a fixed plane. 

3. A Transformation Fixing the Singularities 

The method we propose to apply consists in introducing co-ordinates in such a way 
that the singularities for binary collisions would be fixed, i.e. would be represented 
by fixed values of some of the co-ordinates. In fact, they will be mapped on three 
points equally spaced on the equator of a sphere and the regularization will be 
achieved by mapping these three points conformally on the three vertices of a 
spherical triangle with three right angles. 

The transformation does not imply infinite series, it can be expressed in finite 
terms and the formulae needed may be considered as being relatively simple. 

The connection between the sphere we have just introduced and the three distances 
of the three bodies may be described in the following way. Let y 1; y 2 , y 3 be the 
three angular distances between a general representative point on the sphere and 
the three point? on the equator which represent the binary collisions. Then the three 
distances r k be .ween the bodies are given by 

r k — rV2 sin — . 



210 Regularization of the Three -Body Problem 

More precisely, introducing polar co-ordinates L and B as longitude and latitude 
on the sphere and writing 

we have 

r k 2 = r 2 (l — cos B cos L k ). 

It is obvious that when the distances r k are given these equations permit the 
computation of a kind of mean distance r and of the position of the representative 
point with polar co-ordinates L, B. The representation is twofold, symmetrical 
representative points being equivalent. Equilateral configurations lead to indeter- 
minacies which are precisely those which occur at the pole: B = 7t/2 and L 
indeterminate. 

It is possible to associate with the triangle formed by the three bodies a set of 

rectangular axes such that the six Cartesian co-ordinates of these bodies referred to 

these axes would be 

/2 B L k 
Xk = r A J-cos -cos —, 

/2 . B . L k 

y h = r A / - sin — sin — • 

Uh V 3 2 2 

This can be checked by assuming these values of the co-ordinates and computing 
the mutual distances. 

The values of the masses of the bodies can be defined in a manner similar to the 
foregoing definition of the distances by three quantities m, /j,, v such that the three 
masses m k are 

m k = m 1 + 2/a cos I v — * I 

Then the co-ordinates of the centre of mass referred to the axes just defined are 

x = fir 



y? cos i cos (,+!), 

y?sin|sin(* + ¥ ). 



y = t*r„ s -- 2 



To check this computation it is convenient to obtain and to apply the relation 

2 



Hm k cos — = Smfj, cos I v + —I 



and the corresponding one with sines in place of cosines. We shall need later 
corresponding relations such as 

Hw fc cos L k — 3m/x cos (L — v). 

In order to obtain the kinetic potential (Lagrangian) and the corresj. mding Hamil- 
tonian of the problem, we must consider rectangular axes parallel to the foregoing 
ones but drawn through the centre of mass. The position of these ax* s as regards a 



G. Lemaitbe 211 

set of three fixed rectangular axes will be defined in the ordinary way by the three 
Eulerian angles (cf., e.g., Whittaker, p. 9) and the angular velocity ay*y», will be 
expressed in the usual way.* 

The resulting kinetic energy is expressed in terms of the coefficients of the 
momental ellipsoidf by 

T e = \{Aw 2 + Boy 2 + Ceo 2 - 2Dco x co y ). 

We need also the relative kinetic energy T r and the relative angular momentum 
K r in order to obtain the total kinetic energy 

T = T r + a> z K r + T c . 

Each of these computations requires the calculation, for a, /?, y independent of L, 
summations, such as 

£m fc (a + cos L k + y sin L k ) - Zmp\<x. + cos (L + 2v) + y sin (L + 2v)]. 

k 

They are found to be 

3m(l - /i a )(a + /8r + yA), 

where we write 



and 



T = [u cos (L — v) — fx 2 cos (L + 2v)] 

1 — pr 

A = 5 l> sin (L — v) — fx 2 sin (£ + 2v)\ 

1 — jtt 2 



or, defining n' and v', 

r = p! cos (L — v'), 

A = \x sin (L — v'). 

We write also 

m = m {\ — /u 2 ), m! = m (l — // 2 ). 

We find in this way 



T e = \m Q r 2 
also 



B B 1 

(1 - r)o) x 2 sin 2 - + (1 + T)co y 2 cos 2 - + (1 + T cos B)co 2 - 2 A sin £«y»,J 



m n 



jt r = ^ [(i + r cos 5)r 2 + \r 2 {\ - r cos J B)(Z 2 + £ 2 ) 

— VrrB sin 5 — &rrL cos 5 + %Ar 2 BL sin 5] 
and finally 

j&T, = \m Q r 2 {L sin 5 + BA). 

The kinetic potential (Lagrangian) is then 

L= T + V 



* Whittakek, p. 16, with due allowance for an obvious misprint in the expression of to 2 = <o„ The formula is correct in the 
t Whittaker, p. 124; we write D in place of his F. 



212 Regularization of the Three -Body Problem 

for m k+3 = m k , with 

jj _ V m k+l m k+2 



r k Vl — cos B cos L k 

4. Elimination of the Nodes 

We must proceed to the "elimination of the node" in the classical way (Whittaker, 
p. 344). 

We notice that when the Z axis is taken along the fixed total angular momentum 
K = p we have 

V v = P<p °os Q, Pe = ® 
and, therefore, according to the canonical equations, 

** = <>. 

dd 

It follows that if d is eliminated and replaced by its value in p^ and p^, the modified 
Hamiltonian will still satisfy the canonical equations for the remaining variables. 
The co's can be eliminated by 

Aa> x — Doiy = — p sin 6 cos \p, 

Bco y — Dco x = Py sin 6 sin xp, 

G(o z + K r = p v , 

and this last equation has to be introduced in the expression for the remaining 
canonical momenta. For instance we have 



dB dB dB dB\ 2C / C dB 

It is therefore convenient to write 

= p v dK r =l A. Py , 
B C dB 2 1 + r cos B 
and similarly 

A = Py> dKr = i Pv Sin B 
L C dL 2 1 + r cos B' 

and p B and p L occur only in the expressions 
for which we have 



and similarly 



= Pb 


~A B , 


w L = 


-Pl - 


A L 


6j b = 


d 
dS 


i Tr 


1 

~2C 


K r \ 




a> L = 


a 
dl 


(-. 


1 

~~2C 


Kr 2 ), 




Pr = 


d 

dr 


('•■ 


1 

~~2C 


K*). 





G. Lemaitre 213 



The last steps of the computations may be greatly simplified if we use, in place 
of the variable r, a new variable r , defined by 



Then it is found that 

T r - -^ K 2 = Wo" + I „ , ';'. w .. n x (^ 2 + L 2 cos 2 B) 

reduces to a sum of squares and the transition to'the Hamiltonian is straightforward. 
In this way we obtain finally 



r o = 


= rVl 


+ r cos 


B. 




i 1 


m'r 2 




2" fc 0'0 


1 8 (1 


+ r cos 


BY 



1 



1 p% 



2C m r 2 

with 

T -±k*- l v «i 2(1 + r cos Bf (a*\ ' c » 

and 

p 2 — 7? 2 

7 = | -p^ ^r [-B cos 2 y> -\- A sin 2 ^ — 2D sin ip cos y] 



AB - D 2 



2 ™ 2 



*' // (1 + T cos B) 



2m'r 2 



cos 2 xp sin 2 y> sin 2y 



(! + F ) 5 + (! - T) r - - 2A 



B v ' « sin 5 

sm a — cos 2 — 

2 2 



In £7, r must be replaced by its expression in r . 

5. CONFORMAL TRANSFORMATION 

Now that we have found the Hamiltonian expressed in terms of co-ordinates for 

which the binary collisions are represented by three fixed points on a sphere with polar 

2tt 
co-ordinates B = and L = — k, we must introduce a conformal transformation 

3 

which maps these three points on the three vertices Q v Q 2 , Q 3 of a rectangular triangle. 

We first introduce a complex variable by a stereographic projection 



£ = tang-f)e* 
dB 2 + cos 2 BdL 2 = 4 



and we have 

dm 



(i + a? 



If z is any function of £ and if B L are polar co-ordinates related to z as B and L 
are related to £, then £' being the derivative and tjz the conjugates, we shall have 

dB 2 + cos 2 BdL 2 = i%' |^SJ (dB 2 + cos 2 i^A) 2 ), 
and accordingly we shall have to put in the Hamiltonian 



<b B 2 + 



&l* (1 + il? 



cos 2 B (l + zz) 2 £'£ 



T K + c^j 



214 Regularization of the Three-Body Problem 

where, of course, 

™b = Pb„ — a bj ^i = Pl — A w 
The transformation of the A' a must be such that 

A L dL + AjgdB = A L dL + A B dB . 

The zeros and the poles of the function z correspond to the points which map the 
points B = it 12 (zero) and the antipodal point B = — tt/2 (pole). 

They are: the point, say P , with B Q = it/2 as zero and as poles the points P l5 P 2 , 
P 3 , as symmetrical images of this point relative to the side of the triangle Q v Q 2 , Q s , 
the vertices of which map the singularities of the problem and therefore also of the 
transformation. The points P ', P/, P 2 ', P 3 ' are also poles or zeros. 

The function 

V8 — z 3 

£ = z T= 

i + zWs 

has the required zeros and poles and it will not be difficult to show that the multi- 
plicative constant which is still left undetermined has been chosen correctly. 
The derivative of £ is _ 

V8 — 20z 3 — 2 6 V8 

_ (1 + z 3 V8) 2 

and it can be written 

Vs 

(1 + zV8> L 
where we have written 

VS - 1 ??w 



c = %=- \\(z- c*) (z + -), 

(i + zVs) 2 ,.' Jx \ / v v 



e fc = — 



V2 



for the values of z corresponding to the singular points Q x , Q 2 , Q z . 
We have also 

2tt, 



— ki 
i 3 



1 + 2 3^/8 \ J \ e k I 



and this shows that the points Q fc are really the images of the singularities. 

When M 1 and M 2 are two arbitrary points on the sphere we have for the corres- 
ponding values of z 

. M 1 M 2 (z x - z 2 )(z x - z 2 ) 

sin — . 

2 (1 + z l z i )(l + z 2 z 2 ) 

From this formula we may deduce 



tUi sin 2 MPJ2 
and similarly for the gravitational potential 

G /l A- it 3 . M P k 3 m k+x m k+2 

U = 2V3 - A /-— " fl sin — * . 2 . t Jl. 

r V 1 + zz t'Ji 2 ^=1 sin 2 if # fc 



G. Lemaitre 



215 



This shows that singularities would disappear if the Hamiltonian H were replaced 
by a new Hamiltonian H' ', 

H' = (H - h) sin 2 MQ X sin 2 MQ 2 sin 2 MQ Z = 0, 



to be used with a new time 



/ 



dt 



sin 2 MQ X sin 2 MQ 2 sin 2 MQ 3 



and this gives a regularization of the Three-Body Problem which, besides introducing 
a regularizing time t as in Sundmann's process, introduces also the conformal trans- 
formation and the resulting simplifications of the co-ordinates characteristic of 
Levi-Civita's device of regularization. 

In this paper we have treated the question in a geometrical way. In a preceding 
paper* the question was treated analytically and the computation carried out up to 
the end, including the transformation of the A's and the expression of the non-planar 
terms. 

The classical methods of celestial mechanics are very efficient when the trajectories 
of two of the bodies relative to the third one are well separated, as they are in the 
lunar problem or for the planetary problem with a ratio of the axes not too near to 
unity. These methods could not be applied, for instance, to work out the mutual 
perturbation of planets with encroaching orbits, so that one of them penetrates in the 
inside of the other. The difficulty of such problems is clearly connected with the 
occurrence of the singularity for binary encounter and it may be hoped that it will be 
removed if the regularization process can be sufficiently mastered and used in a 
practical and effective way. 



References 

Happel, H 1941 

Lemaitre, G. . 1952 

Whittaker, E. T 1937 



Das Dreikorperproblem (Koehler, Leipzig). 

"Coordonnees Symetriques dans le Prob- 
lems des Trois Corps", Bulletin de 
V Academic Royale de Belgique (classe 
des sciences), 88, 582-92 and 1218-34. 

Analytical Dynamics (Cambridge Univer- 
sity Press). 



LEMAiTRE (1952). For comparison with the present paper the reader is advised to look first at the formulae (10) and (79). 



Recent Developments in Stellar Dynamics 

G. L. Camm 

Department of Mathematics, University of Manchester 

Stjmmaky 

The dynamical theory of stellar systems, as formulated forty years ago, was apt to be insufficiently 
precise in its treatment of the physical problem. For example, it rarely made use of Poisson's equation 
to relate the star density to the gravitational field in which the stars move. At the same time the theory 
was often too rigid in its mathematical assumptions, as in the adoption of the generalized Maxwellian 
distribution of velocities. In recent years these two points have received much attention from several 
authors. There has also been a growing emphasis on non -steady systems, though this problem has not so 
far yielded any very precise results. The stochastic effects of the gravitational forces between neigh- 
bouring stars have also been considered. 



1. Introduction 

From the beginning of the present century until 1940 the theoretical approach to 
stellar dynamics advanced steadily in a single direction. The statistical approach of 
the kinetic theory of gases was modified to suit the different circumstances, but from 
kinetic theory the Boltzmann equation of continuity and the Maxwellian distribution 
of velocities were carried over and tended to dominate stellar dynamics. The position 
of the theory in 1940 is readily seen in the books of that period by Smart (1938) and 
by Chandrasekhar (1942). Since that time there has been a very noticeable tendency 
to widen the approach, sometimes by introducing a new principle, and sometimes by 
discarding one which had long been accepted. This, then, is a suitable opportunity 
at which to draw attention to the changes that have recently occurred. It is first 
necessary to summarize the fundamental problem, and the older methods of 
attacking it. 

Statistically, the state of a star-system at a time t is described by its velocity- 
distribution function. Suppose (x 1} x 2 , x z ) are cartesian space co-ordinates, and 
(u ly u 2 , u z ) are the corresponding components of velocity. Then the velocity-distri- 
bution function, f(t, X x , X 2 , X 3 , U x , U 2 , U 3 ), is defined in such a way that 
f(t, X x , X 2 , X 3 , U x , U 2 , U^dX 1 dX 2 dX z dU 1 dU 2 dU z is the number of stars for which, 
at time t, X t < x t < X t + dX it U t < u { < V \ + dU it (i = 1, 2, 3). Thus the 
function / can be regarded as the number-density of stars in the six-dimensional 
phase space (X t , U^. 

It should be pointed out that a velocity-distribution function is usually associated 
with a selected group of stars — most frequently with stars of a certain spectral type. 
Different groups may have different distribution functions, but all such functions 
satisfy the same mathematical equation of continuity. It is generally agreed that 
collisions between stars are so infrequent that they may be neglected. On this 
assumption considerations of continuity require the function / to satisfy the Boltz- 
mann differential equation, 

df ^ [ TT df dU; df\ 

a* *-6, 8 l l ^i dt duj 

216 



G. L. Camm 217 

It is usual to suppose that the acceleration, of which the components dUJdt occur in 
this equation, is due to the gravitational field of some smoothed-out distribution of 
matter. 

The actual density of matter in a star system is extremely high in the immediate 
neighbourhood of certain points (corresponding to the stars), and extremely low in 
interstellar space. The gravitational field of such a distribution will obviously have 
large fluctuations in the vicinity of a star, but since close encounters between 
two stars are so rare, it is more convenient to imagine the matter more uniformly 
distributed, so that the density and the gravitational field are smoothed. 

If Q,(t, X t ) is the gravitational potential of this distribution, the fundamental 
equation becomes 

df i df ao an _ 

3* + <=M,3 1 ' *x t + dX, duj " °' 

It was pointed out by Jeans* that this equation simply requires that / is an arbi- 
trary function of the six integrals of the equations of motion, 

dXJdt = U it dUJdi = dQ/dX { (* = 1, 2, 3). 

If Q. is independent of t, then (2D, — U^ — U 2 2 — U 3 2 ) is one of these integrals. If 
D has axial symmetry about the X 3 -axis, there is also an integral of angular momen- 
tum, namely (X 1 U 2 — X 2 U X ). If Q. has spherical symmetry, there are three such 
integrals of angular momentum. Other integrals can only be determined when the 
explicit form of D is given. 

In the case of a steady system possessing complete spherical symmetry the func- 
tion/depends on three variables only, namely the radius, and the radial and trans- 
verse components of velocity. Consequently, there are only two integrals of motion, 
and these are the energy and the square of the angular momentum. The most general 
form of/ for this problem is therefore an arbitrary function of these two integrals. 

In other respects, Jeans's result is rather less helpful. It has frequently been 
supposed that the gravitational field of the galactic system itself has axial symmetry, 
and that the velocity-distribution function depends solely on the energy integral 
and the integral of angular momentum about the axis. On these assumptions, 
the distribution of velocities in planes through the galactic axis should be independent 
of direction. This is not the case in practice. The mean-square of the velocity com- 
ponents in the direction of the galactic centre is significantly greater than that in the 
direction of the galactic pole. 

2. The Ellipsoidal Distribution op Velocities 

An alternative treatment, originally due to Eddington, has been considered in 
great detail by Chandrasekhar (1942). By analogy with the Maxwellian velocity 
distribution in the kinetic theory of gases, it is reasonable to suppose that the 
function / has the form, 

/ = exp {- (aUf + bU 2 * + cU* + 2fU 2 U 3 + 2gU 3 U 1 

+ 2hU 1 U 2 + P U t + qU 2 + rU 3 + s)}, 

where the terms a, b, c, /, g, h, p, q, r, s are functions of X ly X 2 , X 3 and t, to be 
determined. If this form of/ is substituted in the Boltzmann equation, we obtain an 



* Where specific references are not given, the work is reproduced in the books of Smart (1938) and Chandrasekhae (1942) 



218 Recent developments in stellar dynamics 

identity in U^ U 2 and U 3 . For systems in which t is not explicity involved, the 
functional form of the coefficients can readily be obtained. 

In the case of a non-steady system this method has been used by Chandrasekhar 
(1942) to obtain a rather special solution, but no general solution has been found. 

3. The Use of Poisson's Equation 

The outstanding omission in the development of the theory up to 1940, as outlined 
above, was the use of Poisson's equation, connecting the gravitational potential 
with the density distribution which produces it. This was not an oversight, but was 
in recognition of the observed fact that the velocity distribution is not the same for 
all types of star. Yet it is possible to formulate the problem in such a way that 
Poisson's equation can be applied. If the function / is defined so that 

fdX 1 dX 2 dX 3 dU 1 dU 2 dU 3 

is the total mass of the stars which at time t are in the element of phase space 
dX x dX 2 dX 3 dU x dU 2 dU 3 , then / still satisfies the same Boltzmann equation, and in 
addition the integral of/ over all velocities is the mass-density which gives rise to the 
gravitational potential £1. 

The addition of Poisson's equation has provided only disappointing results. It 
has been shown (Camm, 1941) that the steady ellipsoidal distribution which possesses 
the two most obvious properties of the galactic system — namely, a flattened density 
distribution and a rotation about an axis — cannot satisfy Poisson's equation. 

It also appears that the special non-steady solution given by Chandrasekhar is 
unsatisfactory. Schurer (1943) has shown that this exceptional case is directly 
obtained from a steady solution by a change of variables, and Kurth (1949a, 1949b) 
has shown that when Poisson's equation is introduced, this system is either uniform 
in density, or constant in time. 

The case against the ellipsoidal distribution has also been presented by Fricke 
(1951a, 1951b). He has shown that by abandoning this restrictive form, it is possible 
to explain another feature of the observations which the theorists had rather ignored. 
This is the well-known asymmetry of the high- velocity stars. It was first pointed out 
by Oort that the vector velocities of those neighbouring stars whose speeds relative 
to the Sun exceed 63 km per sec. are not distributed symmetrically about the line 
from the Sun towards the galactic centre. Such a state of affairs cannot be explained 
by a quadratic distribution, but Fricke has found that a simple polynomial in the 
two integrals of energy and angular momentum about the galactic axis will serve 
remarkably well as a distribution function for the observed asymmetry. Of course, 
this function still fails to explain the relatively small observed velocity dispersion in 
the direction towards the galactic pole, but it shows that the ellipsoidal distribution 
can usefully be replaced by something slightly more complicated to explain one more 
feature of the observed stellar movements. 

4. Finite Systems 

It is rather remarkable that three quite independent investigations, pursued more 
or less simultaneously in different countries (Camm, 1941, 1950; Fricke, 1951a; 
Kurth, 1949a), all put increased emphasis on the Poisson equation. It is still more 
strange that all should also consider the necessity of finiteness in a mathematical 
model of a stellar system (Camm, 1950, 1952; Fricke, 1951a, 1951b; Kurth, 1949b). 



G. L. Camm 219 

The original generalization of the Maxwellian distribution proposed by Schwarz- 
schild cannot describe a system which is finite in extent. With this model, however 
large a velocity we choose, there is a non-zero number of stars moving with this 
velocity. Thus there is no limit to the kinetic energy of a star, and so the distribution 
in space is not bounded. If, on the other hand, we wish to have a model which is 
finite in extent, then we must ensure that the potential difference between any two 
points of the system is not infinite, and consequently the kinetic energy must have 
a definite upper bound. To obtain a finite system it is therefore necessary — but not 
sufficient — that the form of velocity-distribution funotion should impose a limit on 
the range of velocity components. 

Examples of this kind have been given by Camm (1950, 1952). The aim in both 
these investigations was to find non-steady solutions of the Boltzmann equation 
which were also compatible with Poisson's equation. The systems considered were 
deliberately simplified so as to reduce the number of independent variables. In one 
case, the system was stratified in plane parallel layers, so that only the space- and 
velocity-components normal to the layers were introduced. There is the further 
advantage that the density is simply proportional to the gradient of the gravitational 
intensity. The other simplified model had complete spherical symmetry, so that the 
radius was the only space co-ordinate involved, and there were two components of 
velocity, along and perpendicular to the radius. Again the density and the gravita- 
tional intensity were simply related. 

In each case the mathematical problem could be reduced to a single non-linear 
partial differential equation, but no non-steady solution of either equation has been 
found. Steady solutions of finite radius (or finite thickness in the stratified problem) 
were discovered, and good agreement, especially in view of the over-simplification, 
was found between observed and theoretical density distributions. 

5. Stochastic Forces in Stellar Dynamics 

An entirely new concept was introduced into the theory of stellar dynamics by the 
papers of Chandrasekhar and von Neumann (1942, 1943) on stochastic forces. 
Although stellar encounters are very infrequent in regions of the galaxy such as that 
in which the Sun is situated, they are by no means negligible in their effect. As we 
have explained earlier, it was previously assumed to be sufficient to represent the 
gravitational forces of the whole star system by the gradient of the potential function 
due to a smoothed-out distribution of matter. In this work, Chandrasekhar and 
von Neumann have determined the probability distribution of the gravitational force 
at a point in a field of stars, and also the distribution of the rate-of-change-of -force. 
The application of this work to find the probability distribution of force and rate-of- 
change-of-force on a star moving in the field is bound to be much more difficult, 
since the test-star itself will contribute to the gravitational field, and will alter the 
paths of the field stars. Now the circumstances in which the stochastic forces are 
most important are those rare occasions when a field star passes near the test-star. 
These are precisely the occasions where the present analysis breaks down. In general, 
the test-star will have little effect on the field stars, but for a close encounter the 
effect, both on the path and on the speed, may have remarkable results in changing 
the gravitational field. Obviously the present theory by Chandrasekhar and von 
Neumann overlooks this kind of situation, for the probability distributions do not 
contain any reference to the mass of what we have called the test-star. 



220 Recent developments in stellar dynamics 

In two further papers Chandrasekhar (1943a, 1943b) puts aside the notion of 
stochastic forces due to a field of stars, and considers instead the effect on a test-star 
of a series of binary encounters, developing in this way a theory of dynamical friction. 
Naturally, the masses of the stars enter this kind of treatment. The drawback of 
this method is that the regions where the forces due to random encounters are most 
important are the regions of high density — so that the frequency of encounters is 
raised. But if the density is high, it is extremely difficult to simplify the problem to a 
series of binary encounters. Near the centre of a globular cluster, for example, one 
star may be experiencing fairly large attractions from several neighbouring ones 
simultaneously. 

This work has still to be followed up. The situation with which it deals must 
occur, and will affect the whole velocity distribution, but at the present time the 
theory raises more problems than it solves. 

6. Non-steady Systems 

The various developments in the theory of stellar dynamics which have occurred in 
the past ten years all emphasize the need to consider non-steady systems. A study of 
the observational material points in the same direction. For example, it has already 
been mentioned that different types of stars have very different velocity distributions. 
This cannot be attributed to the constitution of the stars, nor yet to their evolution; 
but it seems likely to be related to the circumstances of their formation, and more 
especially to their age. This is certainly evidence that the velocity distribution is not 
independent of the time. 

There are other reasons for stressing the importance of- non-steady systems. 
We have seen that the effect of binary encounters is not completely negligible ; such 
encounters will certainly produce changes in the velocity distribution, even though 
the process may be extremely slow. Similarly the process of accretion of interstellar 
matter will also cause a slow change. Both these effects have been overlooked in 
formulating the fundamental equation. In a non-steady solution, the motion and 
smoothed gravitational field may produce changes in the velocity distribution which 
are far more rapid than those caused by encounters or accretion, and these may then 
reasonably be neglected. But in so-called steady solutions, these slow changes are 
the only ones which occur, and they cannot therefore be disregarded. 

The search for non-steady solutions is really just beginning. The example given 
by Chandrasekhar proved, as mentioned earlier, to be a trivial one, with no real 
bearing on the general problem. Camm's treatment also revealed a trivial solution, 
but it may be possible to find more general solutions by the methods he has used. 

Von der Pahlen (1947) and Kurth (1951b) have considered the evolution of a 
spherical system, by expressing the distribution function as a power series in the 
time. Kurth (1952) has examined in great detail the mathematical principles involved 
in the general problem, and has shown that in theory a solution can be obtained by 
successive approximations. 

The possibility of periodic solutions has yet to be examined. Since the total energy 
of the system is conserved, an oscillating system seems to be quite feasible. It might 
well be considered with the aid of the virial theorem. Since this theorem is based on 
the equations of motion of each star and the inverse square law of gravitation, it 
seems likely to be directly deducible from the Boltzmann and Poisson equations, 
but this does not seem to have been considered. Clearly the virial theorem gives 



Paul Bourgeois 



221 



much less information than the combination of those two equations, but it so reduces 
the complexities that there may be some advantage gained. It has been used in 
special problems with some success by Freundlich (1945, 1947) and by Kurth 
(1950, 1951a). 

The whole subject, it will be seen, has undergone a remarkable change in recent 
years. The change has not removed the inherent difficulties (and, indeed, we have 
only recently become aware of some of those difficulties), but many new lines of 
attack have been introduced, and progress has certainly been made. Not least 
important is the revival of interest in a branch of mathematical astronomy which for 
twenty years had appeared moribund. 



References 

Camm, G. L 1941 

1950 
1952 

Chandrasekhar, S . . 1942 

1943a 
1943b 

Chandrasekhar, S. and von Neumann, J. . . 1942 

1943 

Freundlich, E. F 1945 

1947 

FRiCKis, W 1951a 

1951b 

Kurth, R 1949a 

1949b 

1950 

1951a 

1951b 

1952 

Pahlen, E. von dee 1947 

Schurer, M 1943 

Smart, W. M 1938 



M.N., 101, 195. 
M.N., 110, 305. 
M.N., 112, 155. 
The Principles of Stellar Dynamics. 

(Chicago). 
Ap. J., 97, 255. 
Ap. J., 97, 263. 
Ap. J., 95, 489. 
Ap. J., 97, 1. 
M.N., 105, 237. 
M.N., 107, 268. 
Naturwissenschaften, 19, 438. 
Astron. Nachr., 280, 193. 
Z. Astrophys., 26, 100. 
Ibid., 26, 168. 
Ibid., 28, 60. 
Ibid., 29, 26. 
Ibid., 29, 33. 
Ibid., 30, 213. 
Z. Astrophys., 24, 68. 
Astron. Nachr., 273, 230. 
Stellar Dynamics. (Cambridge.) 



"Intrinsic" Studies of Stellar Movements in the Milky Way 

Paul Bourgeois 

Observatoire Royal de Belgique, Uccle-Bruxelles 



1. Introduction 

Stellar statistics provides evidence of systematic stellar movements in the Milky 
Way. Stellar dynamics, based on this knowledge, furnishes a simple hypothetical 
basis on which the laws governing the structure and evolution of the galactic system 
may be deduced from observations (Coutrez, 1949, 1951). 

The work of Lindblad and the demonstration by Oort of the differential rotation 
of the Milky Way, shed new light on a series of investigations which had been going 
on since the end of the last century ; in particular, the well-known phenomena of 



222 "Intrinsic" studies of stellar movements in the Milky Way 

Stromberg's asymmetrical pattern of stellar motions and those of Kapteyn's 
preferential motions have thus been explained. 

In the absence of sufficient observational data, attempts to discover the statistical 
laws governing systematic motions within the galaxy were generally based on the 
components of the individual stellar movements, i.e. on proper motions and radial 
velocities. Space velocities were used only in exceptional cases. In these investiga- 
tions use was made of simplifying hypotheses, such as the adoption a priori of a 
Gaussian distribution of velocities. Although such hypotheses are capable of repre- 
senting the observations in a first approximation, they tend to obscure local effects, 
such as the deviations from the galactic differential rotation, which one would logically 
expect in a system as complex as the Milky Way. We know further that the distribu- 
tion of stars in space is not as uniform as one might suppose. The existence of stellar 
agglomerations and of clustering in the dispersive track again show (Ambarzumian, 
1949 ; Blauuw, 1952) that there are deviations from the general pattern of differen- 
tial rotation. In this connection we may also mention the X-effect discovered at the 
beginning of this century. 

2. Method 
Following a study of stellar dynamics by Coutrez (1944), which led to the interpreta- 
tion of the classical isT-effect as the radial components of a spatial if-effect in spiral 
nebulae, we have been engaged in the last few years in a more extensive study of 
stellar motions in space. A general synopsis of work which we have published on this 
subject has been given by Bourgeois (1951). We should like to complete it here and 
to discuss the character and possibilities of the method used, without dwelling on the 
details of the results obtained so far. 

The principle of this method consists of an intrinsic statistical analysis of groups 
of stars of substantially the same mass, the spatial movements of which are well 
known and homogeneous ; the term intrinsic implies that in the course of the analysis 
no use is made of any numerical data not derived from the catalogue itself. Although 
this mode of procedure involves a certain loss of information in consequence of the 
a priori rejection of previously obtained data such as the coefficient of differential 
rotation, it assures great reliability in the analysis of the phenomena studied, just 
because it does not depend on anything outside the system of observational results 
taken as a basis. 

In the course of the investigation the statistics of individual space velocities, of 
barycentric space velocities, and of barycentric space velocity residuals were obtained 
in succession. These different stages permit : (1) the study of the systematic motions 
peculiar to each group; (2) the study of the barycentric velocity field in the 
neighbourhood of the Sun; (3) the study of the peculiar relationship between each 
group and the general field of stellar velocities in the neighbourhood of the Sun. 

3. Observational Data 

The success of such an investigation depends essentially on the homogeneity in 
radial velocity, proper motion, and parallax of the catalogues used in the determina- 
tion of space velocities. Thus no heterogeneous element can be kept in the system ; 
but very refined criteria must be applied to avoid any systematic effect that might 
result from the elimination of some of the stars. This method may reduce the number 
of usable stars, but it increases the accuracy of the results and their representative 



Paul Bourgeois 



223 



value. One may judge this by calculating well-known systematic effects depending 
on a single quantity, such as the classical K-eSect (radial velocities) and the differen- 
tial rotation of the galaxy (proper motions) ; one should then obtain the same result 
from a study of the space motions. Another important indication of the representa- 
tive value of the method can be obtained by comparing the results obtained in 
passing from one system of homogeneous data to another; for example, for proper 
motions, from the system of the General Catalogue to that of the FK3 (Bourgeois 
and Coutrez, 1948a). The homogeneity in mass of the stars in a group is checked 
by the homogeneity in intrinsic luminosity, assuming the mass -luminosity relation. 
Finally, we always attempted to make an estimate of the errors in the calculated 
values. 

4. Individual Space Velocities 

The statistical study of individual space velocities is made by analysing the distribu- 
tion of the representative points of the stars in velocity-space. This distribution 
contains the effect of all the systematic movements of the group. The distribution is 




Fig. 1. A-type stars 
Intersection of the surfaces of equal point-density in velocity-space with a plane perpendi- 
cular to the direction of the galactic centre. The axes of the diagrams are the mean axis 
and the minor axis of the representative ellipsoid, graduated in km per sec. The figures 
on the lines of equal density are numbers of stars per (km per sec.) 3 . The upper diagram 
refers to a representation by a distribution of Charlier's type A, the lower one to an 
ellipsoidal representation. The difference between these two descriptions is very marked. 



224 "Intrinsic" studies of stellar movements in the Milky Way 

represented by a frequency function, from which a series of surfaces of equal point 
density is deduced. A study of these surfaces then permits an easy analysis of the 
distribution. In the work discussed here we have adopted a frequency function of 
Charlier's type A, and we have calculated the coefficients in such a way as to keep 
the moments of the distribution up to the fourth order. A simple Gaussian frequency 
function (Schwarzschild's ellipsoidal distribution) is insufficient, as may be seen from 
an examination of Fig. 1, which shows the deformation in the surfaces of equal 
density that occurs when one passes from a simple Gaussian function to a function of 
this type, characterized by coefficients of asymmetry (skewness) and excess. 

We insist in particular on the necessity for numerical checking in the course of the 
reductions, since the mechanical procedures which are indispensable in such work are 
open to the danger of computational errors. 

To judge the representative value of the frequency distribution obtained, there is 
no question of actually calculating the differences between theoretical and observed 
frequencies. This difficulty is circumvented by successive projection of the observed 
"statistical cloud" on the three do-ordinate axes; the three frequency curves thus 
obtained are then compared with the three theoretical curves deduced from the 
representative frequency function. 

Since series of type A tend towards a Gaussian density distribution when the 
coefficients of order higher than 2 tend to zero, the value of these coefficients here 
gives us a measure of the extent to which the evolution of the system studied is 
affected by random influences. This tendency can be described (Bourgeois and 
Coutrez, 1948b) by the expression: 

D« = 1 + S(a Wfc )», 
where <x. ijk are the coefficients of asymmetry and excess. This expression tends to 
unity when the effective velocity distribution approaches the Gaussian form; but, 
it should be mentioned that the evolution of a group of stars is here considered from a 
statistical point of view in a very small region of the galaxy and may thus present 
peculiar characteristics in relation to the whole. Hence D is not necessarily an index 
of the dynamic evolution of the system. Further, a comparison between the D values 
obtained for stellar groups with different homogeneous characteristics should be 
regarded with caution. 

5. "Barycentric" Space Velocities 

The study of the stellar velocity field in the neighbourhood of the Sun is facilitated 
if individual stellar velocities are replaced by "barycentric" mean velocities of groups 
of stars regularly distributed around the Sun, i.e. the mean velocities with respect to 
the centre of gravity of the groups, thus reducing the effect of individual fluctuations. 
The classical relations used in calculating the constants for the differential rotation 
of the galaxy are : 

u= -xV + y{U-B), 
v= + x(U + B)+yV, 
w = 0, 
where u, v, w are the galactic components of barycentric velocities; x, y, z are the 
galactic co-ordinates of the centres of gravity of the regions ; and where U and V 
are given by U = A cos 2l c and V = A sin 2l c , (l c = longitude of the galactic centre). 
These relations imply a priori a certain form for the coefficients, depending on the 



Paul Bourgeois 



225 



hypotheses used (i.e. mean movements of groups of stars parallel to the galactic 
plane, and rotating about the centre with an angular velocity which depends uniquely 
on the distance from the centre). It is interesting to express these equations in a 
general form of the type 



u. 



= ^a ij x i (i,j = 1, 2, 3), 

so that on the average the velocities u t of the stars in the neighbourhood of the Sun 
depend in the simplest possible manner on their galactic co-ordinates x t (Bourgeois 
and Cotjtrez, 1950b). As no restrictive hypothesis is made here concerning the 




y/fL 


hi 


T T 

o\ 


* 


-"« 


\v 




/nj \ 










IOO \ 








*^ / jS / ± /^ 










* -- "^^/ ^^7 v^ J 


50 \ 








■*■ — 7 L^-^^i j/ / 












•<- / ""W ^^^J**^ i j 


















l~~l — 




^T\~ .1 *^ 


200-l50^1ipoT-~5bL 






J50- 




-I00|j50_200 


•»— — A_ \ — -4^ '2> 












:: 4rV^N3 














^50 / 

\iqo/ 








V YVY 


\vl / 

I5o7 

WLl 






7/ 


► *v 




4 


4 





Fig. 2. A-type stars 
Representation of the field of barycentric velocities of deformation. The heavy curves 
are line of flow, the sense of motion being shown by the arrows. The lighter curves are the 
intersections of surfaces on which the velocity has a given value, respectively by the galactic 
plane (below) and by the plane normal to the galactic plane and to the direction of the 
centre (above). The oij'-axis is directed towards the galactic centre. The axes are graduated 

in parsecs, and the curves of constant velocity in km per sec. 
16 



226 



"Intrinsic" studies of stellar movements in the Milky Way 



coefficients a w the latter appear as the components of a tensor which represents the 
linear relation (valid in the first approximation) between the velocities and the 
co-ordinates. These components represent the group's local rotation and local 
deformation, the latter giving rise to a linear dilatation and to a shear. By calculating 
this tensor a„ one may obtain a precise idea of stellar movements in the vicinity of 
the Sun in the linear approximation. In particular, one may judge the extent to 
which the stars in a group which has been studied follow the classical differential 
rotation (Bourgeois and Coutrbz, 1950a, 1950b). Fig. 2 illustrates, for A-type 
stars, the characteristics of the velocities of deformation in the local group, taken with 
respect to the centre of gravity of the system. 

6. Residual Space Velocities in the Barycentric System 

Another method of studying the special characteristics of the stellar velocity field in 
the neighbourhood of the Sun consists in separating the barycentric space velocities 




Fig. 3. B-type stars 
Relation between residual barycentric velocities (spatial -fiT-effect), distances, and longi- 
tudes, represented in the galactic plane. The vectors have a marked tendency to align 
themselves parallel to the direction from the Sun to the galactic centre, thus indicating the 
tendency of the galaxy, taken as a whole, to show an expansion effect. The diameters of 
the points give the accuracy. The full-line vectors were calculated on the basis of the 
FK3 system, those shown by broken lines used the system of the General Catalogue. It 
will be noticed that the results are essentially unaffected by transition from one system 

to the other. 

from the effects of the apex and of the classical differential rotation. In this way one 
obtains the barycentric velocity residuals as a function of longitude and distance. 
Fig. 3, obtained from a study of stars of spectral type B, illustrates the distribution 
in the galactic plane. 



Paul Bourgeois 



227 



Similarly, one can separate the barycentric space velocities from the general effects 
of a linear dependence on the co-ordinates as discussed in Section 5. The barycentric 
residual space velocities found in these two cases are closely correlated (Bourgeois 
and Coutrez, 1950a), which shows that the linear approximation used above gives a 
very satisfactory description of the region of space studied, without the use of any 
special hypothesis. 

7. Conclusions 

The results obtained in the course of the studies which we have pursued in this field, 
in collaboration with A. Coutrez, are concerned with stars of spectral types B and A 
(Bourgeois, 1951). 

For these spectral types the concentration of the representative points near the 
centre of gravity in velocity-space has turned out to be particularly strong, and it 
may be attributed to the presence of streaming, probably in a spiral arm of the 
Milky Way. The elongation of the "distribution clouds" in a direction close to that 
of the galactic centre and their flattening normal to the galactic plane are a strong 
indication of the presence of differential galactic rotation, and follow from the position 
of the Sun in our stellar system. 

The method described in Section 5 reveals a local rotation of the two groups studied 
about axes inclined to the galactic plane ; this seems to be due to turbulence in the 
neighbourhood of the Sun. 

The study of barycentric residual velocities for stars of spectral type B shows, for 
the groups studied, a tendency to radial expansion parallel to the galactic plane and 
to concentration towards this plane. 

These few examples are given here merely to indicate the interest of this method, 
which is very promising for the detailed study of stellar movements in the vicinity 
of the Sun. 

Recent work in radio astronomy promises to provide knowledge of the Milky 
Way in depth. The radial velocities obtained from observations of the radiation from 
interstellar hydrogen at 21 cm, related to the distances through the adoption of a 
scheme for the motion of the entire system, already provide a proof of the spiral 
character of the galaxy in agreement with Morgan's model. 

Progress in stellar dynamics depends essentially on an ever-increasing knowledge 
of the structure of our galaxy. The development of complementary investigations 
of this kind is the only means of securing this. Such studies should therefore be 
actively pursued. 



References 



Ambarzumian, V. A 1949 

1950 
Biaauw, A 1952 

Bourgeois, P 1951 



Bourgeois, P. and Coutrez, R. 



1948a 



"Associations stellaires", Astron. J. U.S.S.R., 

26, 3. 
"Les associations stellaires et Forigine des 

etoiles", Acad. Sci. U.S.S.B., 14, No. 1, 15. 
"The Age and Evolution of the £ Persei Group of 

O- and B-type Stars", Bull. Astron. Inst. 

Netherlands, 9, 433. 
"Les mouvements spatiaux des etoiles dans 

l'etude de la Galaxie", del et Terre, 1-2; 

Commun. de VObservatoire Royal de Belgique, 

No. 24 (with additional references). 
"Comparison des composantes tangentielles des 

vitesses spatiales residuelles dans les systemes 

du General Catalogue et du FK3 pour les 



228 



The K effect in stellar motions 



Cotjtrez, R. 1944 



1949 



etoiles de type spectral B", Ann. Astrophys., 
11, fasc. 3; also Commun. de I'Observatoire 
Royal de Belgique, No. 7. 

1948b "Etude statistique intrinseque des etoiles de 
type spectral A a mouvement spatial connu", 
Ann. de VObservatoire Royal de Belgique, 
3e Serie, tome III, fasc. 4, p. 209. 

1950a "Mouvements systematiques a 1'approximation 
lineaire et vitesses spatiales r^siduelles pour 
les etoiles de type spectral B", Ann. Astrophys., 
tome 13, No. 2; also Commun. de VObservatoire 
Royal de Belgique, No. 17. 

1950b "Mouvements systematiques a 1'approximation 
lineaire et vitesses spatiales residuelles pour 
les etoiles de type spectral A," Hie Congres 
National des Sciences, Gompte-rendus et 
Commun. de I'Observatoire Royal de Belgique, 
No. 18. 
"La dynamique des nebuleuses spirales", Ann. 
de VObservatoire Royal de Belgique, 3e serie, 
Tome III, pp. 76-80. 
"Contribution a l'etude de la Dynamique de* 
systemes stellaires", Ann. de VObservatoire 
Royal de Belgique, 3e Serie, Tome IV, fasc. 3. 

1951 "Methodes actuelles de la dynamique stellaire 
appliquees aux problemes de la Voie Lactee et 
des nebuleuses extragalactiques", Scientia, 
January; also Commun. de VObservatoire 
Royal de Belgique, No. 22. 



The K-Effect in Stellar Motions 

Harold F. Weaver 

University of California, Leuschner Observatory, Berkeley, U.S.A. 
Summary 

The large K term associated with bright early B stars signifies expansion, but does not indicate any 
fundamental characteristic of the local region of the galaxy. It is a chance property of the sample popula- 
tion traceable to nearby community motion groups within the sample. Removal of chance and real 
community motion groups reduces the K term to approximately + 2-5 km per sec. for the O to B2 stars. 
This term may be attributed principally, if not entirely, to gravitational red shift. 

A method of investigating on the basis of galactic structure the significance of the negative K term 
found for faint B stars is suggested. Tests indicate : 

(1) No variation of the K term within the local spiral arm of the galaxy. 

(2) No relative motion of the local and inner spiral arms, hence the same K term holds for both. 

(3) While stars in the outer arm also tend to confirm the normal K term, the stars in the Perseus- 
Cassiopeia region possess large negative peculiar radial motions. Inclusion of the numerous B stars in 
this region will bias any solution towards a small or negative K term and a large Oort .4 -constant. 



1. Discovery of the ./^-Effect 

Fifty years ago Frost and Adams (1904), from a discussion of twenty measured 
radial velocities corrected for solar motion, discovered the two principal characteristics 
of the motions of the B stars. 

( 1 ) The average peculiar motion of the B stars is small, approximately 7 km per sec. 

(2) The average residual with regard to sign is positive and unexpectedly large, 
-f- 4-6 km per sec, according to Frost and Adams' pioneering investigation. 



Harold F. Weaver 



229 



In 1910 these findings were confirmed by Kapteyn and Frost (1910) and simul- 
taneously by Campbell (1911, 1913), who had available radial velocities of 138 B 
stars reasonably well distributed over the sky. Campbell found the average value of 
the difference observed radial velocity minus computed solar motion component to 
be -f- 4-93 km per sec. To represent this clearly present but unexplained systematic 
motion of the B stars, Campbell introduced into the usual solar motion equation an 
empirical constant K, 

v r — K — u Q cos a cos d — v Q sin a cos d — w Q sin d. .... (1) 

Here v r represents the expected radial velocity of a star situated at position <x, 6 ; 
u Q , v Q , w q represent the components of solar motion in a rectangular co-ordinate 
system oriented in the customary manner in the a, <5 frame. Thus introduced, a 
non-zero K term indicates only that the average velocities of B stars in opposite 
parts of the sky are not equal in numerical value. 

2. Observational Characteristics of the if -Term 

The K term is large for O and early B stars ; for later spectral types it is small or 
negligible. Numerical values of the K term found by Campbell and Moore (1928), 
and more recently by Smart and Green (1936), are shown in Table 1. 

Table 1. Numerical values of the K term 



Spectral 


Campbell and Moore 


Smart and Green 










type 


K term 


Number 


K term 


Number 




(km per sec.) 


stars 


(km per sec.) 


stars 


B 


+ 4-9 


284 


+ 4-7 


645 


A 


+ 1-7 


500 


00 


742 


F 


+ 0-3 


199 


-0-6 


523 


G 


- 0-2 


244 


- 10 


433 


K 


+ 0-3. 


687 


-0-2 


1118 


M 


+ 0-7 


234 


00 


222 



The numerical value of the K term is a function of the apparent magnitude range 
of the B stars investigated. This correlation, first pointed out by J. S. Plaskett 
(1930), is illustrated by the data of Table 2. 



Table 2. Variation of K with apparent magnitude 
(Data from J. S. Plaskett, 1930) 



Type 


Magnitude 
range 


Average 
magnitude 


Number 
of stars 


K 

(km per sec.) 


0-B2~ 
0-B2 


< 5-5 
> 5-5 


3-98 
6-60 


78 
139 


+ 4-3 
+ 01 



3. Interpretations of the if -Effect 

Various explanations of the K term have been suggested. 

(1) Erroneous Wavelengths or Unknown Blends of Lines Used for Radial Velocity 
Determinations. Campbell (1911, 1913) pointed out that an average increase of only 
0-07 A in the wavelengths of all fines would eliminate the K term. Errors of this size 
do not exist in modern wavelength scales, yet the K term persists ; the explanation 
fails. The triplet series of He is composed of very closely spaced lines which are 



230 The K-effect in stellar motions 

blended on small dispersion spectra. Modern data do not support the early suggestion 
that the individual line intensities are abnormal in stellar spectra and hence that the 
wavelengths of the blends used for radial velocity determination are incorrect. 

(2) Pressure Shift of Spectral Lines. A few decades ago the principal source of line 
broadening in B stars was thought to be pressure, which was also assumed to shift 
the lines longward (Campbell, 1911, 1913). To-day it is clear that the pressure in 
B-star atmospheres cannot produce the broad lines or the K effect. 

(3) Atmospheric Currents. In 1914 Campbell, basing his remarks on Evershed's 
measurements of convection in the solar atmosphere, suggested that much greater 
descending currents in the atmospheres of B stars might account for the K term. 
It is difficult entirely to eliminate this suggestion on the basis of scanty modern data. 

(4) Expansion of the Galaxy. Pilowski (1931), Ogrodnikoff (1932), Milne 
(1935), and earlier investigators have suggested that the K term represents a real 
expansion of the entire galaxy or of the local cluster in which the Sun is presumably 
located. Expansion is generally taken to imply that K should increase with distance 
from the Sun. Failure to find an increase of K with an increase of apparent magni- 
tude of the stars observed is evidence against the interpretation. It does not rule 
out expansion of the local cluster. 

(5) Relativity Red Shift. Freundlich (1915) attributed the K effect to the gravita- 
tional red shift of spectral lines in the massive O and B stars. Calculations indicate 
(Plaskett, 1930; Plaskett and Pearce, 1936) that for 0-B2 stars the relativity K 
term is between 2 and 3 km per sec. While the relativity red shift is undoubtedly 
present in the B stars, it cannot account for the entire K term shown in Table 1 . 

(6) Stream Motions and Moving Clusters. Plaskett and Pearce (1929) suggested 
that the K term can be traced to groups of stars in community motion. In particular, 
they asserted that removal of the Scorpio-Centaurus Stream from the data decreases 
the K term to the amount to be expected from relativity shift. 

Smart (1936) has criticized the particular explanation and certain specific question- 
able mathematical procedures used by Plaskett and Pearce. In spite of these 
specific criticisms, the original suggestion of moving clusters loses none of its 
applicability. Suitably generalized, it offers the most promising means of explaining 
the effect. 

4. Generalization of the Moving Cluster Concept 

(1) Real or Chance Community Motion Croups. Solar motion solutions made with 
equation (1) contain the implicit assumption that the stars surveyed constitute a 
homogeneous sample population in random motion. A departure from random motion 
in any subvolume of the sample implies that the stars in the subvolume possess com- 
munity motion. No physical association of the involved stars is necessarily implied 
by such community motion, which may be solely a chance phenomenon. 

If an appreciable fraction of the sample stars possesses community motion, the 
precise values of the derived K term and solar motion will not necessarily have 
fundamental significance. They will represent only properties of the sample popula- 
tion studied, not fundamental properties of the region of the galaxy surveyed. If 
the stellar population as a whole is inhomogeneous and various subpopulations possess 
community motion, then, as the observed sample is varied and community motion 
groups are gained or lost, the numerical value of the K term and the solar motion 
may be expected to change. 



Harold F. Weaver 231 

Percentagewise, the change in the K term may be large. The numerical value of 
the K term is determined by : 

(a) physical properties of the stars (M/R, for example, determines the gravita- 
tional red shift), and 

(b) the state of motion of the sample population. 

In this later connection, the K term indicates whether the sample population is, on 
the average, expanding, contracting, or stationary. B-star samples are generally 
small ; a few stars in community motion will give the sample the property of expanding 
or contracting and will greatly influence the value of K. 

(2) Fictitious Community Motion Groups. Galactic rotation imposes a field of 
differential motions in the sample volume. The stars in any sub volume of the sample 
space will, in general, have group motion with respect to the stars in any other sub- 
volume. To allow for the effects of galactic rotation, the equation for solar motion is 
usually written 

v r — K — u Q cos a cos d — v Q sin a cos d — w Q sin d -\- rA sin 2(1 — Z ) cos 2 b, 

....(2) 

where the symbols have their customary definitions. Equation (2) is valid only if 
f is not correlated with I orb; its use implies that the stars are distributed uniformly 
in space, or for the B stars, at least distributed uniformly in the galactic plane. 
Further, it may be noted that the galactic rotation term in (2) is valid only for nearby 
stars, though it is often applied to rather distant ones. A more precise equation is 
desirable. The expression (see Trumpler and Weaver, 1953) 

v r = K —u Q cos a cos 6 — v Q sin a cos d — w Q sin d + R [a)(R) — co ] . sin (I — l ) cos b 

....(3) 

is preferable. For circular galactic orbits the rotation term of (3) is exact. 

Equation (3) applies to an object at distance R from the galactic centre; w(R) 
represents the angular velocity of a body moving in a circular orbit of radius R ; 
R is the value of R for the Sun; a> Q represents co(R ). As in equation (2) we assume 
that the object under discussion is so close to the galactic plane that <o(R, z) may be 
taken as co(R, 0) or simply co(R). For B stars the assumption is adequate. 

Very early B stars (B2 and earlier) sharply define the spiral arms of the galaxy 
(Morgan, Whitpord, Code, 1953; Weaver, 1953a). Stars of later spectra type 
appear to delineate the arms much less clearly. The principal arm structures denned 
by stars B2 and earlier in the solar neighbourhood are indicated in Fig. 1. The distri- 
bution of the early-type stars within the arms is very spotty; major distribution 
trends are indicated by shading in Fig. 1. Use of equation (2) requiring no correlation 
of r and I is clearly inadmissible for the B2 and earlier stars, particularly when great 
distances are involved. Use of equation (2) will introduce fictitious star drifts in those 
directions in which the average distance differs from the f value used in the equation. 
These fictitious drifts will distort the results for solar motion and may introduce a 
spurious K term, the numerical value of which will depend upon the space distribu- 
tion of the stars. If equation (3) is employed in the discussion, and if each star is 
treated individually for galactic rotation, space distribution irregularities and approxi- 
mations in the galactic rotation term of (2) will not affect the results. However, 



232 



The K-effect in stellar motions 




Fig. 1. Schematic representation of principal spiral structures in the solar neighbourhood. 
The Sun is indicated by the circle; the Scorpio-Centaurus aggregate is shown as a dotted area. 

star groups of real or chance community motion will still influence the values 
derived. Such localized effects must be detected and treated individually. 



5. Examples of iT-TERM and Solar Motion Solutions 

We consider principally stars B2 and earlier and magnitude 5-0 and brighter. Radial 
velocities are available for eighty stars satisfying these criteria ; nearly all lie within 
a few degrees of the galactic plane. For purposes of exhibiting velocity results 
graphically, all stars will be treated as though they were in the plane. No loss of 
detail or accuracy will occur as a result, only a small additional fictitious scatter will 
be introduced in the graphical representation. All calculations are carried out pre- 
cisely; no such approximation is made. The galactic concentration of the B stars 
will permit only a very weak determination of B , the latitude of the apex. However, 
it will play no part in the discussion. 

(1) Fig. 2 shows the eighty stars projected on the galactic plane. Fig. 3 exhibits 
the observed radial velocity, v r , of the eighty stars plotted as a function of galactic 
longitude, I. Solar motion appears in this diagram as a single sine wave, galactic 
rotation as a double sine wave of at most a few kilometres per second amplitude, 
since the stars are nearby. A K term representing a general expansion or gravitational 
red shift appears as a constant and raises the entire group of points. Community 
motion groups appear as distortions of these three features and need follow no regular 
pattern. 



0.8 
06 
0.4 
0.2 



235 c 



0.2 
0.4 
0.6 



Harold F. Weaver 
145° 



233 



1 


1 


1 ■ 1 

• 


1 


1 


1 


1 


1 




- 




• 














* 




• 














- 




*. • 


• 








• 


- 


— - 


• 


• • + 


• 

• 
• 


• 


• 
• 


• 




— 


1 


1 


, . -1 


1 


1 


1 


1 


1 





55° 



10 0.8 0.6 0.4 02 02 0.4 0.6 0.8 

325° 



12 



Fig. 2. Galactic plane distribution of 0-B2 stars for which radial velocities are 
available. Unfilled circles represent members of Scorpio -Centaurus aggregate 



(i) Solar motion, galactic rotation, and K term do not permit a satisfactory repre- 
sentation of the observed points in Fig. 3, which define a distorted, asymmetric 
curve. The points lie predominately above the zero axis ; they attain a maximum at 
approximately I = 220°, a minimum at I — 15°. The curve requires 205° to rise, 
155° to fall. 




325 



0° golactic long. 90° 180° 270° 

Fig. 3. Observed v r , I - relation for stars shown in Fig. 2 



360° 



(ii) The points to the left and to the right of I = 195° show a difference in disper- 
sion. The reader may demonstrate this for himself by covering first one half of the 
diagram and then the other : the points to the left of 195° represent stars lying within 
the local arm pictured in Fig. 1 ; the points to the right represent stars outside the 
local arm in the region occupied by the Scorpio-Centaurus group. The distortion of 
the v r , I relation appears to be caused primarily by the points to the right of 195°. 



234 



The K-effect in stellar motions 



Discussed on the basis of equation (1), the points of Fig. 3 lead to the parameter 
values 

K = + 4-75 km per sec, S Q = 19-46 km per sec, L Q = 16?2, B Q = + 9?3. 

(2) The points shown in Fig. 3 may be individually corrected for galactic rotation: 

v r ' = v r — E [a)(B) — Q) ] sin (I — l ) cos b 



K 



u Q cos a cos d 



v^ sin a cos 6 — w„ sin d. 



•(4) 



For this calculation R has been taken as 8-75 kiloparsecs; the function a)(R) — co 
has been determined by Weaver (1954) on the basis of radial velocities and the 




0° galactic long. 90 



Fig. 4. Observed v r ', I relation for stars shown in Fig. 2; Scorpio -Centaurus members 
omitted. The curve represents solution number 5 

revised distance scale of the cepheids. The modifications caused by galactic rotation 
are slight since the stars are all nearby. The peak at I = 220° is slightly lowered. 
The points in the range I = 235° to 325° are raised; this effect increases the 
asymmetry. The radial velocities v r ' discussed by equation (4), lead to the results 

K = + 4-88 km per sec, 8 Q — 19-37 km per sec, L Q = 16?8, B Q = + 9?3. 

Correcting for galactic rotation has slightly increased the K term, decreased the solar 
motion, and shifted the apex to a greater longitude. These effects can be traced 
principally to the raising of the points in the Z-range 235° to 325° and to the fact that 
there are essentially no early B stars in the quadrant 325° to 55° to balance the 
change. 

(3) The disturbance in the solar motion curve arises mainly in longitude range 
235° to 325°. Thirteen Scorpio-Centaurus stars identified in this /-range by Blaauw 
(1946) are plotted in Fig. 2 as open circles. We remove these objects from the data 
and find for the remaining stars 

K= + 3-35 km per sec, S Q = 18-73 km per sec, L Q = 9?5, B Q = + 14?6. 

The Scorpio-Centaurus stars thus account for 1-5 km per sec. of the K term. 

A plot of the v r \ I relation of the sixty-seven stars to which the solution applies 
and the representation afforded by the solution are shown in Fig. 4. 



Harold F. Weaver 



235 



(4) The most pronounced difference between the plotted points and the curve in 
Fig. 4 occurs in the longitude range 190° to 235°, where the points he predominantly 
above the computed curve. If the slightly large velocity found for these stars is of 
significance in galactic motions, we should expect to obtain the same result from other 
B stars in the same direction and region of space. If the same high velocity is not 
found from these other stars, we shall be inclined to judge that the bright B stars 
form a community motion group. 

In Fig. 5 (a) the solid dots show the galactic plane distribution of the bright B stars 
for which the v r ', I relation is plotted in Fig. 4. Fainter, generally more distant early 



(a) 



(b) 



(c) 



1 


"T - - 


1 




~T 








"1 ' 

145° 


- 










» 


• 




- 


- 235° 




°°° 


• 


.. 


• 


• 


** * 
*..t* 


+sun 


i 


" 


1 




...... 1 




1 




325° 

i 





1 




♦ 50 


"1 




* : 




♦ 40 


" 




. • 




♦30 


; 




# 




♦ 20 


PS. " 






^>~ 


+ 10 



-10 


. 


















• 


" 


i 


_ 


-20 
-7n 


. 



'2 I 165° 215' 265' 165" 215° 265* 

Fig. 5 (a). Galactic plane distribution of O— B2 stars in longitude interval 195°— 235°. 
Filled circles, stars brighter than magnitude 5-0 

(b) The v r ', I relation for stars represented as filled circles in Fig. 5 (a). Curve is same as 

that drawn in Fig. 4 

(c) The v r ', I relation for stars represented as unfilled circles in Fig. 5 (a). Curve is same 

as that drawn in Fig. 4 

B stars for which velocities are available are shown as open circles. Fig. 5 (b) exhibits 
the v r ', I relation for the stars indicated by solid dots in Fig. 5 (a). The curve drawn 
is a section of that shown in Fig. 4. Fig. 5 (c) shows the v r ', I relation for the stars 
represented by open circles in Fig. 5 (a) and repeats the curve of Fig. 5 (6). 

(i) The B stars pictured in Fig. 5 (c) do not exhibit high velocities as do the stars 
in Fig. 5 (6). We take this to indicate that the stars pictured in Fig. 5 (b) form a 
chance community motion group. 

(ii) The stars outside the local arm in the 235° direction show the large velocity 
dispersion characteristic of stars in the inner arm of the galaxy (Weaver, 1953b) or 
of some of the stars in the region of the Scorpio-Centaurus group. 

(5) In the sample population treated in solution (3) we replace the stars for which 
we found chance community motion (Fig. 5 (&)) by the stars shown in Fig. 5 (c). 
We find for this altered sample that does not contain the chance community motion 
group 

K = + 2-89 km per sec, S Q = 18-94 km per sec, L Q = 9?9, B Q = + 22?0. 
The K term has been further diminished ; the solar speed is not significantly changed. 

(6) The disturbance in the v r ' , I relation in Fig. 4 appears to arise primarily from 
those stars outside the local arm. If we make use only of those stars in the local arm 
(the stars in solution (5) minus those in the Z-range 235° to 0°) we find for the forty 
stars available. 

K = 2-07 km per sec, 8 Q = 19-71 km per sec, L Q = 8?0, B Q = + 23-3. 



236 The K-effect in stellar motions 

The K term is still further diminished. We note that the solution is not a strong one, 
however, since only 235° of longitude are covered. Nevertheless, the trend is clear : 
the local-arm stars show a small If -term. 

6. The Faint B Stars 

We investigate the decrease in the K term with increasing magnitude by considering 
two related questions. 

(a) Do distant early B stars located in the local arm show, after they have been 
individually freed from the effects of galactic rotation, the same v r ' , I relation as the 
nearby local-arm stars ? 

(b) Do early B stars located in the inner and outer arms indicate, after they have 
been individually freed from the effects of galactic rotation, the same v r ', I relation 
as the nearby local-arm stars ? 

(1) To investigate the first problem we make use of a number of early B stars and 
galactic clusters of type 1B2 or earlier, distributed in the galactic plane as shown in 
Fig. 6 (a). Radial velocities for the clusters have been kindly supplied by Dr. R. J. 
Trumpler. The v r ', I relation derived for the distant B stars and clusters is illustrated 
in Fig. 6 (6), where the results of solution (5) are shown as a curve. Though the 
scatter in the v r ' , I relation found for these more distant objects is somewhat larger 
than that found for the nearby B stars (owing to distance uncertainties and hence 
galactic rotation uncertainties) visual inspection of the diagram indicates no signifi- 
cant differences in the results obtained from the nearby and distant local-arm stars. 
The stars within the local arm thus appear consistent in their motions. 

(2) Fig. 6 (a) also shows projected on the galactic plane several galactic clusters 
(1B2 or earlier in type) and individual B stars located in the inner and outer arms. 
The v r ', I relation derived from these is shown in Fig. 6 (b). 

Inner Arm. Visual inspection discloses no discrepancy between the v r ', I relation 
of the inner arm and that found for the local arm. Examination of the inner arm is 
made generally difficult by the large velocity dispersion found for stars in that arm. 
Significant values can be derived only if observations are available for a considerable 
number of stars. 

Outer Arm. The v r ' , I relation of that portion of the outer arm studied agrees 
generally with the v r ', I relation of the local-arm stars. The Double Cluster in Perseus 
(and all field stars in the region of the Double Cluster) are notable exceptions to this 
rule. That the abnormal velocity of the rich stellar association in the Per-Cas region 
represents community motion rather than a direct effect of galactic rotation can be 
shown by comparing the value of a>(R) — (o (see equation (4)) derived from local- 
arm stars with that derived from Per-Cas stars of the same R. We find 
[(o(R) — ft> ] Per . Cas ^ l-5[co(R) — eo ] local arm ; the abnormal velocity of the stars in 
the Per-Cas region is of the order of — 20 km per sec. 

Community motion in the Per-Cas region is not confined to B stars; it is also 
shown by the cepheids (Weaver, 1954). 

Inclusion of the extensive Per-Cas community motion group in the determination 
of the K term, solar motion, or galactic rotation constant will strongly bias the results. 
Velocities of many Per-Cas stars are included among the data discussed by Plaskett 
and Pearce ; these Per-Cas stars have decreased the K term and increased the Oort 
constant A. However, discussion of such details is beyond the scope of the present 
paper; they will be considered in a more extensive discussion elsewhere. 



Harold F. Weaver 
145° 



237 



235° — 




325 c 



(a) 



+ 40 — 




325° 



galactic long. 90 



(b) 



Fig. 6 (a). Galactic plane distribution of 0-B2 stars and clusters. Filled circles: members 
of local arm; unfilled circles: members of inner arm; crosses: members of outer arm. 
Large symbols represent clusters; small symbols represent individual stars. The shaded 
area represents the location of thirty-seven stars treated as one point in the discussion. The 
large cross represents the double cluster in Perseus 
(6) The v/, I relation for objects pictured in Fig. 6 (a). Symbols as in Fig. 6 (a). 



7. Conclusions 

The large K term associated with the bright early B stars is real and signifies expan- 
sion, but does not indicate any fundamental characteristic of the local region of the 
galaxy. It is a chance property of the sample population traceable to nearby com- 
munity motion groups within the sample. The largest of the community motion 
groups is the Scorpio-Centaurus cluster. Its influence on the K term was first pointed 



238 



The K-effect in stellar motions 



out by Plaskbtt and Pearce. When community motion groups are removed from 
the data, a K term of approximately +2-5 km per sec. remains for the O to B2 
stars. This may be attributed principally if not entirely to gravitational red shift. 
A method of investigating on the basis of galactic structure the significance of the 
small K term found for faint B stars is suggested. Brief, incomplete tests that can 
be made with available data indicate : 

(1) Motions within the local arm are consistent; there is therefore no variation of 
K term within the arm. 

(2) No relative motion of the local and inner arms is found, hence the same K term 
holds for both arms. 

(3) While stars in the outer arm also tend to confirm the normal K term, the stars 
in the Per-Cas region possess large negative peculiar motions. Inclusion of the 
numerous B stars in this region will bias any solution towards a small K term and a 
large Oort constant. 

Though it is not discussed in detail, there is mentioned the danger of introducing 
fictitious community motion groups among distant stars by making inadequate 
corrections for galactic rotation. Inspection of former investigations shows that at 
times this has been a powerful source of bias. 



References 
Blaauw, A 1946 

Campbell, W. W 1911 

1913 

1914 

Campbell, W. W. and Moore, J. H 1928 

Finlay-Freundlich, E 1915 

Frost, E. B. and Adams, W. S 1904 

Kapteyn, J. C. and Frost, E. B 1910 

Milne, E. A 1935 

Morgan, W. W., Whitpord, A. E. and Code, A. D. 1953 

Ogrodnekoff, K 1932 

Pilowski, K 1931 

Plaskett, J. S 1930 

Plaskett, J. S. and Pearce, J. A 1929 

1936 

Smart, W. M 1936 

Smart, W. M. and Green, H. E 1936 

Trumpler, R. J. and Weaver, H. F 1953 

Weaver, H. F 1953a 

1953b 
1954 



Publ. Kapteyn Astron. Lab., Groningen, 
No. 52. 

Lick Obs. Bull., 6, 101, 125. 

Stellar Motions (Yale University Press, 
New Haven). 

Lick Obs. Bull., 8, 82. 

Publ. Lick. Obs., 16, xxxviii. 

Astron. Nachr., 202, 17. 

Dec. Publ. Univ. of Chicago, 8, 105. 

Ap. J., 32, 83. 

M.N., 95, 560. 

Ap. J., 118, 318. 

Z. Astrophys., 4, 90. 

Z. Astrophys., 3, 53, 279, 291. 

M.N., 90, 616, 621. 

Amer. Astron. Soc, 6, 277. 

Publ. Dominion Astrophys. Obs. Victoria, 
5, 227 (gives an excellent historical 
account of the problem of the K-term). 

M.N., 96, 568. 

M.N., 96, 471. 

Statistical Astronomy (University of Cali- 
fornia Press, Berkeley), Chapter 6.21. 

Astron. J., 58, 177. 

Publ. Astron. Soc. Pacific, 65, 132. 

Astron. J., 59, 375. 



On the Empirical Foundation of the General Theory of Relativity 

E. Finlay-Freundlich 

University Observatory, St. Andrews, Scotland 



1. Introduction 

I first had the pleasure of meeting Professor Stratton at an eclipse expedition 
in Sumatra twenty-eight years ago. On the occasion of writing this summary of 
my eclipse work, I hope he will share my pleasant recollection of the time we spent 
there together. 

During the past years the task of putting the general theory of relativity to the 
test has receded into the background; and thus among many scientists the feeling 
seems to prevail that the theory has been successfully tested, or that the astronomical 
observations have not yet reached the necessary accuracy to decide definitely whether 
the new effects predicted by the general theory of relativity exist or not. I need only 
refer to P. G. Bergmann, 1946, p. 221.* 

Both attitudes are wrong. Since various new attempts are in progress to settle 
this problem, which is of fundamental importance, it is justified to summarize the 
present situation. 

One of the three possibilities of testing the general theory of relativity refers to 
the orbital motion of a small mass in the gravitational field of the Sun ; the improved 
law of motion, as compared with Kepler's law, predicts an advance of the perihelion 
of Mercury moving in the gravitational field of the Sun alone to the amount of 
42?9 per century. This effect is, as such, not of a fundamentally new character but 
a refinement of Newton's law of gravitation by the theory of relativity. The latest 
exhaustive discussion of all observational material, covering the time from 1765 to 
1937, by G. M. Clemence, Washington, results in a complete agreement between 
theory and observation. As to the planet Venus, the eccentricity of its orbit is too 
small to allow a sufficiently accurate determination of the motion of the perihelion 
of its orbit, and for the still more distant planets the predicted effect becomes 
too small. 

The two other possibilities of testing the theory, the red-shift of the solar fines and 
the deflection of light passing near the Sun, concern fundamentally new phenomena, 
namely an influence of a gravitational field upon the propagation of electromagnetic 
energy. These two predictions of the theory of relativity have not yet been confirmed ; 
not, however, because the predicted effects are too small to be determined with the 
necessary accuracy. In both cases, as will be shown, a determination of the predicted 
effect is possible with a standard deviation of not more than about 5 per cent of the 
predicted value. Both effects either dp not agree with the theoretical prediction, or 
else they are strongly falsified by secondary effects of unknown origin and character. 



* See also the Notes in the last paragraph of this article (p. 246). 

239 



240 



On the empirical foundation of the General Theory of Relativity 



2. General Red -shift of the Solak Lines 

Let us first consider the case of the general red-shift of the solar lines. 

Compared with the wavelength of the corresponding lines, produced in a terrestrial 
light source, all solar lines taken from any point of the surface of the Sun, allowing, 
naturally, for the influence of the Sun's rotation, should appear displaced towards 
the red end of the spectrum by an amount AX equal to + 2- 12 x 10~ rt times the wave- 
length. This shift corresponds to a loss of the energy A(hv) of the light quanta when 
rising from the Sun to the Earth's surface, equal to their gain in potential energy in 
the gravitational field of the Sun. Fig. 1 reproduces the results of the latest observa- 
tions by M. G. Adam (1948) of Oxford, compared with observations made many years 

ro" 3 A 



|8 

I 

6 
5 



OO Ol 0-2 CK5 0-4 05 0-6 07 0-8 0-9 hO 
SIN<9 — — »■ 

Fig. 1. Comparison of wavelength changes across the disk of the Sun 
• = Oxford (Miss Adam, 1948), A6100, 14 lines, mean intensity 6 
o = Potsdam (Freundlich et al.), A4420, 9 lines, mean intensity 3, mean of 12 radii. 

earlier in Potsdam. The latter observations were only relative measurements of wave- 
length changes along twelve radii on the Sun's disk ; for the centre of the Sun the 
observations of Oxford and of Potsdam are made to coincide. All lines should reveal a 
shift equal to the constant amount 12-9 X 10 -3 A represented in Fig. 1 by the parallel 
fine. The probable error of each measurement represented by a black dot in the graph 
is about 10 per cent. The complete coincidence of the two independent sets of obser- 
vations shows clearly that the red-shift is not just "barely observable" ; nor is it justi- 
fied to add "but appears to be in agreement" with the theoretical predicted value. 
Fig. 1 shows quite clearly that, if the general red-shift were existent according to the 
theoretical prediction, the observations would reveal it with absolute certainty; 
what they reveal, however, is (1) that indeed a certain general red-shift is indicated 
(for the values do not drop at the Sun's centre below + 5 x 10 -3 A), but they defi- 
nitely do not indicate a constant red-shift of the amount 2-12 x 10 -6 X A; (2) near 
to the Sun's limb the red-shift rises suddenly, steeply to a value which seems to lie 
near to the predicted effect. 

The possibility exists therefore that the theoretically predicted effect is existent, 
but that it is eclipsed by other effects. Disturbing effects that may be made 





1 

PREDIC1 


ED R 


ED-S 


HIFT 


















































































































































f 
JJ 












» 


--.« 


'0 





























E. Finlay-Fkeundlich 



241 



responsible are Doppler effects due to radial currents in the solar atmosphere and 
pressure effects in the solar atmosphere. 

Many attempts have been made to bring the observational results into agreement 
with the theory by taking account of both possibilities ; but neither an acceptable 
model for radial currents in the solar atmosphere could be constructed, nor is there a 
generally accepted theory of pressure effects which make it possible to extract a 
constant red-shift of the demanded amount from the observational data. 

So far, the marked increase of the red-shift near to the Sun's limb is the only effect 
firmly established. At the limb Doppler effects due to radial currents will not affect 
the wavelengths of the solar lines ; similarly, the influence of pressure effects should 
become insignificant near the limb when the lines arise from the highest layers of the 
solar atmosphere. 

Therefore, observations restricted to the Sun's limb should yield a reliable proof 
for the existence of the predicted red-shift, if the observed red-shift converged at the 
limb towards the theoretically predicted value. This, however, according to the 
present results, is definitely not the case. 

Evershed (1936) has obtained extensive material using lines from the reversing 
layer near to the limb of the Sun, as well as from the chromosphere and from erup- 
tions rising well above the Sun's limb. The results of these observations are very 
consistent, but they yield a red-shift at the Sun's limb considerably larger than the 
predicted value. The following table gives all necessary data concerning these 
observations by Evershed (1936). 

Table 1 



AA bs 

O limb minus 

vacuum arc, 

Mt. Wilson 



O limb minus 
vacuum arc, 
Evershed 



Excess over 

relativistic 

effect 



Number of lines and 
spectra used 



'Fe lines: 
3932 A 
(max. of 14 lines) 
Reversing Layer < 4433 A 

(max. of 11 lines) 
6203 A 
l (max. of 5 lines) 
Chromosphere : H and K lines . 

Prominences: H and K lines 



+ 0-0141 A 
+ 0-0148 A 
+ 0-0248 A 



+ 00146 A 
+ 00139 A 
+ 0-0252 A 
+ 00151 A 
+ 0-0150 A 



+ 00063 
(62 per cent) 

+ 0-0045 
(50 per cent) 

+ 0-0120 
(90 per cent) 

+ 0-0068 
(80 per cent) 

+ 0-0067 
(79 per cent) 



14 lines 
50 spectra 
11 lines 
25 spectra 

5 lines 
25 spectra 

2 lines 
22 E. and W. spectra 

2 lines 
180 spectra 



The obvious disagreement with the theoretical prediction led to the ad hoc hypo- 
thesis of an additional "limb effect". Since no explanation for such a limb effect can 
be given, it is more consistent to summarize the results as follows : 

At the Sun's limb, where neither radial currents nor pressure effects in the atmosphere 
of the Sun will seriously affect the wavelengths of solar lines, a red-shift is observed which, 
however, does not have the value predicted by the theory of relativity, but a value con- 
siderably larger than the one predicted by it. 

Frequently the results obtained from observations of the companion of Sirius are 
cited as a reliable confirmation of the theory of relativity. These observations 
definitely do indicate the existence of a red-shift ; the value may even roughly agree 
with the predicted amount ; this is by the way also the case with the solar observa- 
tions ; for the average red shift which would result if one derived the mean value by 



242 On the empirical foundation of the General Theory of Relativity- 

integrating over the whole solar disc, would not fall far short of the predicted value. 
The possibility of a detailed investigation along the Sun's disc reveals, however, that 
the problem is not so simple. The experimental proof of the theory of relativity 
asks for more than a crude confirmation of the theoretical prediction. The results 
which can be derived from observation of Sirius B are far too uncertain to yield a 
sound foundation for an empirical test of the theory of relativity. 

3. The Light-Deflection 

The situation with regard to the deflection of a light beam passing near to the Sun is 
very different from that concerning the general red shift discussed in the preceding 
paragraph. While in the latter case the existence of a general constant red shift of 
the solar lines relative to terrestrial lines has not yet been safely established, the 
existence of a light deflection of the expected order, decreasing apparently propor- 
tional to 1/A (A denoting the smallest distance at which the beam passes the Sun) 
is beyond doubt. It remains undecided, however, whether the observed and predicted 
values of the deflection agree with each other. With the increasing accuracy of the 
determinations of the light deflection, a systematic difference between observations 
and theory has been disclosed with increasing certainty, the observed values exceeding 
the theoretical prediction by nearly 25 per cent. The whole effect exceeds considerably 
the observational errors; it is therefore quite misleading to say that it "is just 
outside the limits of experimental error" (Bergmann, 1946, page 221). The standard 
deviation of a determination of the light deflection need not exceed 5 per. cent of the 
predicted value. The difficulties in measuring accurately the light deflection are 
partly of a practical, partly of a theoretical nature. 

The practical difficulty in measuring the light deflection with sufficient accuracy 
lies in the fact that the observations have to be performed under the exceptional 
conditions during the short and rare moments of a total solar eclipse. The duration 
of the last total eclipse in Sweden in 1954, for instance, was only about 150 sec. 
The theoretical difficulty lies in the fact that the method of deriving accurate 
stellar positions from astrographic plates cannot be applied to this problem, unless 
special precautions are taken. This fact was not recognized at one time and has 
reduced the weight of the earlier determinations of the light deflection. Table 2 
gives the five determinations for which the mean error does not exceed i 0*3. 
The aim should be to determine the fight deflection with an accuracy of i 0"1 or 
less; according to the theory the value should be equal to 1''75 for light passing 
the Sun's limb. Observations made in Potsdam with the powerful apparatus 
specially constructed for this problem, and made under normal night conditions, 
show that it should be possible to bring the standard deviation even below the 
value of ± 0''1 ; v. Brttnn and v. Kluber (1937). 

Apart from the result of the Lick expedition, given in the second line of Table 2, 
all values exceed considerably the predicted value of 1''75; also in the case of the 
Lick expedition the value actually observed, before having been specially corrected, 
was equal to 2" 05. 

The only method which promises to yield the value of the light deflection with the 
necessary accuracy is a differential comparison of the star field surrounding the 
eclipsed Sun during a total eclipse with the same field under normal night conditions. 
On the eclipse plate the stars should appear shifted away from the Sun's centre when 
compared with their position on the night plate. Chosing for a numerical comparison 



E. Finlay-Freundlich 243 

the proportions of the Horizontal Twin Camera of 850 cm focal length, specially 
devised for this problem, a star the light from which grazes the Sun's limb should 
appear shifted by 0-07 mm. In practice, stars will hardly ever be visible through 
the solar corona as near as that to the Sun ; the actually observable shifts will not 
surpass 0-03-0-04 mm. If, however, all stars in a field of 3° X 3° numbering more 
than, say, 20, show a systematic general shift of this amount, this shift is determinable 
with great certainty ; and if the reduction is correctly performed, i.e. with complete 
knowledge of the theoretical implications, the value for the deflection at the limb 
can be derived with a standard deviation not exceeding 5 per cent of the theoretically 
predicted value. 

By using, as an intermediate reference plate, one exposed through the glass to the 
star field surrounding the Sun during the eclipse, the positions of all stars visible 
on an eclipse plate and on a night plate can be compared with each other, so 
to speak, emulsion in contact with emulsion; and relative co-ordinates Ax,, Ay, 
can be measured. The two values Ax, and Ay, for every star i (i — 1, 2, 3, . . . 
n, if n is the total number of stars on the plate) arise from various sources and can 
be expressed by the equations : 

Ax, = Zx + O .y,+ 8 . x, + px, 2 + qx,y, + L— + Bx,+ Ax, 

' i 

Ay, = Zy-0.x i + S.y i + px,y, + qy* + L^ \ + By, + Ay,. 

Here the x„ y, are the rectangular co-ordinates of the star i relative to the centre of 
the Sun which is supposed to coincide with the centre of the plate ; r, is the distance 
of the star from the Sun's centre ; Zx and Zy denote the small relative shifts of the 
centres of the two plates, one eclipse and one night exposure, when brought into 
contact and measured in the comparator ; O is a small difference in their orientation 
relative to the reference system of co-ordinates ; 8 denotes the unavoidable difference 
of scale on both plates. Due to the fact that, since the exposures of the night sky have 
to be made either a few months before or after the eclipse, these exposures are made 
under different conditions as to temperature, etc., and the focal length of the telescope 
will not have been accurately the same at both exposures. The coefficients p and q 
account for possible small differences in the inclination of the photographic plates in 
the plate-holder relative to the optical axis of the telescope during their exposures. 
L represents the expected influence of the light deflection on the positions of the stars 
on the eclipse plate ; and finally A and B the influences of aberration and refraction 
upon the relative positions of the stars ; both corrections naturally change with the 
atmospheric situation and the conditions at the time of the exposures. 

The last two effects can be accurately determined, if the time of the exposure and 
if the temperature and barometric conditions during the exposure are noted. These 
effects can consequently be calculated and subtracted, before formulating the above 
equations. Similarly, in order to simplify the following discussion, we shall drop the 
terms with p and q, for in rigidly constructed telescopes of steel the position of the 
plates relative to the optical axis can be kept accurately under control. 

The remaining 2n equations have to be solved for the five unknowns Zx, Zy, 0, 
8, L. If 2n ^> 5 a least squares solution should yield accurately their most prob- 
able values. This, however, is only true if the determinant of this set of equations does 



244 On the empirical foundation of the General Theory of Relativity 

not vanish ; when this is the case, two of the unknowns must be linearly coupled and 
one of them can be freely chosen. It is a special misfortune for the problem under 
consideration that this singular case really endangers the accurate determination of 
the two unknowns L and 8. For a difference in scale of two plates that are compared, 
will, assuming always the Sun's centre to coincide with the plate's centre, increase in 
a linear manner in all directions away from the centre. The light deflection on the, 
other hand, is supposed to decrease proportionally to the distance from the Sun; 
that means along a hyperbola. But in practice the hyperbola is indistinguishable 
from its asymptote. In the area from 4-10 solar radii from the Sun — this is the 
region in which the majority of stars lie that contribute measurably to the light 
deflection — the hyperbola differs from the linear decline by no more than ^- 0"04; 
the mean error of one co-ordinate measurement is about eight to ten times this 
amount. Thus L and 8 are quasi-linearly coupled. That means L can only be 
determined accurately if 8 is independently determined by a special set of observa- 
tions and not introduced into the above set of equations. 

The weight of the determination of L is given by the expression 



W-, 



\h a) 



if L and 8 are both derived from the least squares solution; the weight is, however, 

tt> 2 — w . -, if # is determined independently. These two expressions refer to the 

simple case that the stars are symmetrically distributed around the Sun, the centre 
of which is supposed to coincide with the centre of the plate ; h denotes the harmonic 
mean, a the arithmetic mean of the r f 2 . The second value w 2 reveals the unfavourable 
conditions under which L has always to be derived. This is due to the fact that the 
value of the light deflection decreases with r t ; and since the majority of the stars lie 
in the outer region of the plate and consequently do not contribute to the deter- 
mination of L with sufficiently large, accurately measurable, shifts the light deflec- 
tion cannot be derived from a least squares solution as an unknown of large intrinsic 
weight. The few stars lying near to the Sun will always be the backbone of the 
determination. If, however, 8 is left in addition to L, to be derived from the least 
squares solution, the weight of the determination of L is further seriously reduced 



by the smallness of the factor I 1 • 

J \h a) 



Thus L can only be accurately determined if 8 is derived by special independent 
observations, and only if the determination of 8 itself is extremely accurate. For 
any error made in the determination of 8, say AS, will produce, due to the tight 
coupling of L and 8, an error AL in the determination of L, equal to AL = — h AS ; 
here the possible effect of an asymmetrical distribution of the stars around the Sun is 
neglected. Since h is of the order of f t 2 , and f t rarely is smaller than 4-5 solar radii, 
an error in the determination of 8, producing an error of the assumed value for the 
Sun's radius equal to 0''01, or, expressed in other terms, an error of the scale corres- 
ponding to no more than 1/100,000 of the focal length of the telescope, will increase 
or reduce the value obtained for L by 0''2. 

So far only in the case of the Potsdam expedition of 1929 was the scale correction 
independently derived with sufficient accuracy to reduce the mean error of L to 



E. Finlay-Fkeundlich 



245 



i 0"1. That this accuracy was really attained is proved by the fact that observa- 
tions with a twin telescope, during the eclipse, of a second star field so distant from 
the Sun that no influence of a light deflection could be expected — this field being 
reduced in the same way as the eclipse field — gave, as they should, no shifts of the star 
positions. The mean value of the measured changes of star positions, in this second 
field, taken irrespective of sign, did not exceed 0'-13. 

The two preceding determinations of L (see Table 2) are not based on independent 
observations to control S ; of the two later determinations the last one failed in its 
attempt to derive S independently, while of the Russian expedition no detailed 
results are available. 

Table 2 



Expedition 


L 


Standard 
deviation 


Remarks 


Greenwich, 1919 . 
Lick, 1922 . 
Potsdam, 1929 . 
Moscow, 1936 
Yerkes, 1947 


1*98 
1*72 
2*24 
2*71 
2*01 


± 0*16 
± 0*15 
± 0*10 
± 0*26 
± 0*27 


A revised calculation gives 2*16 

Original, uncorrected, result : 2*05 

Scale correction derived by special observations 

Scale correction not derived independently 



However, all these determinations point towards a larger value of the light 
deflection; they obviously do not scatter around the predicted value of L = 1''75, 
but around a value of the order of 2-2.* 

To summarize : while the prediction of the general theory of relativity, concerning 
the more accurate law of the orbital motion of a small body around the Sun, appears 
to be in full agreement with the observational results, the predictions of effects of a 
fundamentally new character, concerning the motion of photons in the gravitational 
field of the Sun, have not yet been safely confirmed. The existence of a general, 
systematic red-shift of all solar fines, relative to the corresponding terrestrial lines, 
amounting to 2-12 x 10 -6 times X has not yet been established. The observations 
indicate a small red-shift of the solar lines, but the value changes across the disk of 
the Sun, apparently affected by superimposed effects of a different character. At the 
Sun's limb, however, where the disturbing influences of Doppler effects and pressure 
effects should become insignificant, the observed red-shift does not converge towards 
the value predicted by the theory of relativity. As to the light deflection, the 
existence of it is absolutely safely established. Its quantitative value, however, is 
not yet ascertained. Observations point towards a value nearly 25 per cent larger 
than the theoretically predicted one. The increasing accuracy of the determinations 
does not tend to reduce this disagreement, but on the contrary, seems to bring it 
more clearly to light. 

Both effects are well beyond the limits of observational errors, so that research 
should be once more concentrated on a final solution of these problems. A repetition 
of the experiment to measure the light deflection with the powerful special telescope 
developed in Potsdam is in preparation. The Deutsche Akademie der Wissenschaften 
had generously given the necessary funds to improve this instrument, so that in case 
of favourable conditions of observation it should be possible to redetermine the value 



* In this connection reference should be made to a recent result obtained by G. van Bibsbroeck, in Astron. J., 58, 87-88, 
1953, who derives a value of 0?70 ± 1?10 from his observations at the Khartoum eclipse of 25th February, 1952. See also the 
author's previous discussion with van Bibsbroeck in the Astron. J., 55, 49, 1950; 55, 245, 1951; 55, 247, 1951. 



246 On the empirical foundation of the General Theory of Relativity 

of L with a standard error not exceeding ^ 0*1. The observation of the total 
eclipse of 30th June, 1954, in Oland, Sweden, was prevented by clouds; we are 
now hoping for 1955 in Ceylon. Also new observations concerning the general red 
shift are planned. 

Notes added in proof 

A new hypothesis concerning the origin of the red-shift of spectral lines in stellar 
atmospheres has recently been put forward by the author in the Nachrichten der 
Akademie der Wissenschafien in Oottingen, Mathem.-Phys. Klasse, 1953, No. 7, 
pp. 96-102 (also in Gontrib. Obs. St. Andrews, No. 4); see also the subsequent article 
I.e., pp. 102-8 by Max Born ; furthermore, a note by the author in Proc. Phys. Soc. 
(London), A, 67, pp. 193-4, 1954; and the more detailed paper in Phil. Mag. Ser., 
7, 45, pp. 303-19, 1954 (also in Contrib. Obs. St. Andrews, No. 5). 



References 

Adam, M. G 1948 M.N., 108, 446. 

Bergmann, P. G 1946 Introduction to the Theory of Relativity. Prentice 

Hall, New York. 

Evershed, J 1936 M.N., 96, 152. 

Brunn, v. and Rluber, v 1937 Z.f. Ap., 14, 242. 



SECTION 4 



THEORETICAL ASTROPHYSICS 



Lecturer: "... FundamentaUy, a star is a pretty simple 

structure ..." 
Voice from the Audience: "You would look pretty simple, 

too, at a distance of ten parsecs." 

Colloquium in Cambridge University, 1954. 



Fraunhofer Lines and the Structure of Stellar Atmospheres 

A. Unsold and V. Weidemann 

Institut fur Theoretische Physik und Sternwarte der Universitat, 
Kiel, Germany 



Summary 

A review is given on the development of our ideas concerning stellar atmospheres and their spectra. 
After some brief remarks concerning the early comparisons, between stellar and laboratory spectra it is 
emphasized that quantitative analysis of stellar spectra must involve essentially three points : ( 1 ) spectro- 
photometry measurements; (2) the theory of radiative and possibly convective transfer; and (3) the 
atomic theory of absorption coefficients. After following them for the period 1905-39, the more recent 
developments are characterized by treating radiative transfer in spectral lines mostly according to the 
scheme of "true absorption" and by giving closer attention to the temperature-stratification, using 
essentially non-grey models. Finally, the importance of more and better theoretical, and particularly 
laboratory work on transition probabilities, line broadening, etc., is stressed. The problem of damping 
constants in solar type stars is given special attention. 



1. Direct Comparison of Stellar and Laboratory Spectra 

In the early days of stellar spectroscopy astrophysicists mostly attempted to interpret 
the spectra of celestial bodies by comparing them directly with those of laboratory 
sources. That procedure was very successful as to the comparison of wavelengths 
and identification of elements. Also, it happened to lead to a correct interpretation 
of sunspot spectra in terms of lowered temperature while, for instance, the comparison 
of the spectra of novae with those of underwater-sparks led to quite erroneous ideas. 
Quantitative conclusions could by no means be derived from a direct comparison 
of stellar and laboratory spectra since one essential characteristic of all cosmical 
fight sources can never be reproduced on earth : namely their enormous dimensions. 

2. Quantitative Interpretation of Stellar Spectra in Connection 

with Spectrophotometry Measurements, Theory of Radiative 

Transfer and the Atomistic Theory of Absorption Coefficients 

Quantitative interpretation of stellar spectra became possible only by connecting 
spectrophotometric measurements on solar and stellar spectra with the increasing 
knowledge of atomic physics through the intermediary of theory. The latter in this 
connection has a twofold function : it must consider (a) the transfer of energy through 
stellar atmospheres by radiation and possibly convection, and (6) the atomic theory 
of absorption and re-emission coefficients for continuous and line radiation under the 
conditions prevailing in stellar atmospheres. Let us recall briefly the essential 
steps on these two lines of research up to about 1939. 

(a) Radiative transfer and radiative equilibrium* 

After some preliminary work by Schuster and others the theory of radiative transfer 
and radiative equilibrium was established in two fundamental papers (1905 and 1914) 



* We propose to speak of radiative transfer quite generally, when energy is transported as radiation, of radiative equilibrium 
however when all the energy flux is carried by radiation only. 

.249 



250 Fraunhofer lines and the structure of stellar atmospheres 

by K. Schwarzschild. In the second paper he fully realized the importance of 
spectrophotometric measurements on line profiles and briefly indicated also that of 
the theory of line absorption. Most of Schwaezschild 's further work was connected 
with Boer's theory of spectral lines, and he would certainly have returned to the 
subject of Fraunhofer lines had not an untimely death prevented him from doing so. 
Ten years later the Bohr theory was firmly established and in 1924 Russell and 
Stewart reviewed various mechanisms that might contribute towards the broadening 
of stellar absorption lines. In 1927 Unsold tried to connect specially made spectro- 
photometric measurements of solar fine profiles with the theory of radiative 
transfer and the quantum theory of fine absorption, in order to determine the electron 
pressure P e and — at least partly — the quantitative chemical composition of the solar 
atmosphere. But let us first follow the further development of the theory of radiative 
transfer. 

In The Internal Constitution of the Stars (1926) Eddington had considered various 
"models", i.e. various assumptions concerning the ratio of line and continuous 
absorption as a function of depth in the atmosphere. But it was only in 1928 that 
Milne pointed out clearly the role of continuous absorption as limiting the depth of 
the layer which contributes towards the formation of Fraunhofer lines.* Practically 
speaking, that meant an important step toward the explanation of the absolute 
magnitude effects which are used for determining spectroscopical parallaxes. A 
general theory of radiative transfer in the case of "weak absorption", i.e. weak lines 
or the wings of strong lines, was established in 1932 by Unsold under the title of 
"theory of weight functions"; the contribution of atoms located at a given depth 
towards the depression in a line being characterized by certain functions of depth 
which can be calculated once for all. Minnaert (1936) rightly remarked that the 
variation with depth should be taken into account accurately not only for the ratio 
of line to continuous absorption but also for the Kirchhoff-Planck-function deter- 
mining the re-emission of radiation. Calculations in that direction were made by 
Minnaert himself and later by Unsold, Kurz, and Hunger (1949-50). Perhaps we 
should note that Eddington's old idea, that the intensity I v (0, &) leaving the solar 
surface at inclination # to the normal should roughly equal the Kirchhoff-Planck- 
function B v for an optical depth t„ = cos #, was put on a firmer mathematical basis 
by Barbier in 1943 and later applied also to fine-problems by Unsold (1948). 

The one-sided emphasis laid in the literature upon radiative transfer should not 
distract us from giving more attention also to problems of convection. After the 
fairly obvious thermodynamical foundations had been given by Unsold in 1930, 
Biermann first established the principles of connective energy transport in stellar 
atmospheres. Progress in the application of these ideas to the actual interpretation 
of spectra is difficult and slow, chiefly owing to aerodynamical difficulties. 

Before going into the modern developments concerning the theory of radiative 
and convective transfer as well as structure of stellar atmospheres we should firsl 
report the earlier history of line absorption and related problems. 

(b) Atomic theory of line absorption and emission 

In his 1927 paper Unsold had still used a semi-classical theory of radiative damping, 

taking into account quantum-theoretical values for the oscillator strength/, but not 

.. * ,f* ^at time many astrophysicists considered as the important point in Milne's papers his "generalized theory of ioniza- 
tion taking into account the increase of pressure with depth. That, however, was" an illusion, since actually the corresponding 
increase of temperature (which had been ignored) is just as important. wne»puuum K 



A. Unsold and V. Weidemann 251 

yet for the damping constants y of the lines. Fortunately for the chiefly considered 
resonance lines, the error was not significant. A correct quantum-mechanical theory 
of radiative damping was given in 1930 by Weisskopf and Wigner. In 1929 
Struve clearly demonstrated that the hydrogen and helium lines of early type 
spectra were broadened by the Stark effect due to electric fields in the ionized 
matter. Minnaert (1931) recognized the importance of the thermal Doppler effect 
and Strtjve and Elvey (1934) that of turbulence. They all used very effectively 
for the analysis of solar and stellar spectra the "curve of growth"-method whose 
history goes back to an important paper by Russell, Adams, and Moore (1928). 
Minnaert's finding of unexpectedly large damping constants and his observation 
(with Genard) on the strange appearance of the diffuse Mg-series in the Sun finally led 
Unsold in 1936 to apply also the quantum theory of collision damping to astrophysics. 

Here we should interrupt our report for a moment to recall that in 1938 our 
quantitative ideas concerning the pressure and the hydrogen-abundance in later type 
atmospheres underwent a considerable change through Wildt's remarkable discovery 
of the continuous absorption by the H~ ion, followed by the well-known papers of 
Stromgren and Chandrasekhar. 

About 1939 most workers in the field of stellar atmospheres had the impression 
that we had essentially the necessary fundamental knowledge for going into a detailed 
and "final" analysis of solar and stellar spectra. But it was just work of that type 
that led to an almost complete reversal of our ideas during the second world war. 

3. Modern Development of the Theory of Radiative Transfer; 
Local Thermodynamical Equilibrium; Structure of Non-grey 

Stellar Atmospheres 

Hitherto the radiative transfer of line radiation had been mostly treated following 
K. Schwarzschild's scheme of "scattering" or "monochromatic radiative equili- 
brium", using Milne's terminology. In 1942, Houtgast in connection with the 
interpretation of his observations on "Variations in the profiles of strong Fraunhofer 
lines along a radius of the solar disc", carefully studied the mechanism of incoherent 
scattering. Houtgast's and Spitzer's work indicated that actually the wings of 
these fines should closely follow Schwarzschild's scheme of "true absorption" or 
"local thermodynamical equilibrium", in Milne's language. At the same time 
Unsold noticed that — judging from quantum-theoretical arguments — practically 
all the lines of early type stars should be formed by "true absorption" and confirmed 
that conclusion by measurements of the central intensities of all the sufficiently 
strong lines. 

To-day we are convinced that — perhaps with the exception of some features in 
the resonance fines of late type stars — the re-emission in practically all Fraunhofer 
lines can be calculated fairly accurately simply by applying Kirchhoff's law. That 
relieves the astrophysicist from a heavy mathematical burden. The transfer problem 
for true absorption then admits the well known exact solution which is so simple that 
even in the most complicated cases it can be handled numerically rather easily. 

However, naturam expellas furca, tamen usque recurret . . . ! Up to 1942 it was 
almost generally agreed that the increase of temperature with depth in stellar 
atmospheres could be calculated with an accuracy sufficient for spectroscopic 
purposes using the simple theory of the "grey" atmosphere in connection with the 
Rosseland average over the absorption coefficient. However, in early type stars the 



252 Fraunhofer lines and the structure of stellar atmospheres 

variation of the continuous absorption coefficient with wavelength amounts to several 
powers of ten near the Lyman-limit of hydrogen and in later type stars the densely 
crowded absorption lines involve huge variations of the absorption coefficient over 
considerable parts of the spectrum. Such deviations from the previously assumed 
greyness lead to a steep decrease of the temperature in the highest layers of the 
atmospheres. For instance, the boundary temperature of the Sun, which had been 
given in 1938 generally as 4830°K is now definitely known to be as low as 3800°K. 

The mathematical theory of radiative equilibrium in non-grey atmospheres is 
very cumbersome indeed. It is easy to write down their general equations, but the 
numerical evaluation is beset with difficulties. A first idea as to their behaviour 
could be derived from a most useful paper, which Chandrasekhar had published as 
early as 1935 on the influence of lines on the temperature stratification of the Sun. 
In 1947 Stromgren and Unsold proposed the so-called "A-iteration" procedure in 
which — starting from a reasonable zero approximation of the source function — one 
calculates first the radiation intensity, then again (taking into account the conserva- 
tion of energy) the source function and so on. This method, however, gives reasonably 
fast convergency only for the topmost layers. Therefore in 1951 Unsold proposed 
the "flux-iteration" procedure. Here, again starting from some zero approximation, 
one calculates exactly the corresponding radiative flux as a function of depth and 
tries then to correct its deviations from constancy by means of a relation between 
source function and flux which had been first published by R. v. d. R. Woolley in 
1941. The second method works well in the deeper layers but is not so good in the 
high ones. So probably the thing to do will be to apply first the flux-iteration and 
then the A-iteration. 

For the deeper layers the models of later type atmospheres must moreover be 
improved by taking into account the convection: Unfortunately, the practically 
most important layers exhibit energy transfer by convection and radiation. It is, 
however, just the transition from the radiative to the convective zone of a stellar 
atmosphere that is theoretically most difficult to handle. Possibly it will be necessary 
also to allow for the fact that in a convection zone one has at one and the same 
optical depth quite a considerable range of temperatures. In the accessible layers of 
the solar atmosphere that range of temperatures connected with the boiling of the 
granulation may be about i 500°K. 

Calculation of model atmospheres on this new standard, of course, takes much time. 
In Kiel we have tried to give the thermodynamical and atomistical fundamentals in 
a series of papers "Der Aufbau der Sternatmospharen". The solar atmosphere has — 
with particular emphasis on convection — been treated by E. Vitense in two papers 
published in Z. Astrophysik. Work on earlier spectral types is being done in con- 
nection with the analysis of large dispersion plates kindly taken for us by O. Struve. 
At other institutes similar problems have been tackled in somewhat different ways ; 
we should mention especially the work done by Stromgren in Copenhagen, de 
Jager (Diss. Utrecht, 1952), J. C. Pecker (Diss. Paris, 1951), etc. 

4. Numerical Values of Continuous and Line Absorption Coefficients 

and Damping Constants, etc., Calculated from Quantum Mechanics and 

Measured in Laboratory Light-sources 

All the work which we have briefly indicated so far can, however, only lead to actual 
extension and higher precision of our knowledge about the stars if it is supplemented 



A. Unsold and V. Weidemann 253 

by a better knowledge of the involved atomistical constants : absorption coefficients, 
oscillator strengths, damping constants, etc. 

Important progress in this direction has been made using quantum mechanical 
methods. Let us recall only the calculation of transition probabilities by Hartree, 
Biermann, Green, Bates, and their collaborators, and the exceedingly laborious 
calculation of the H - absorption coefficient by Chandrasekhar. 

However, many of the astrophysically important atomic or ionic transitions are 
very unsuitable for quantum theoretical calculation. So it will be necessary that 
astrophysics take up again much closer connection with laboratory spectroscopy. 
The aim of such work will not be — as in the early days — to duplicate astrophysical 
spectra in the laboratory, but to measure the atomistic constants entering into the 
theory of solar and stellar spectra. Nevertheless it is necessary to develop light- sources 
approaching more or less stellar conditions. In any case they must have a definite tem- 
perature and often — what is more difficult to realize — also a known electron pressure. 

Oscillator strengths or transition probabilities for many spectra of astrophysical 
interest have been measured in the electric furnace by R. B. and A. S. King (1935) 
in absorption and by Carter (1949) in emission. Absolute values have also been 
determined by Kopfermann and Wessel (1949-51) using an ingenious atomic beam 
method avoiding the use of a vapour pressure curve. However, in the most important 
case of iron the Gottingen and the Pasadena measurements differ by a factor 3 whose 
origin could not yet be traced in spite of considerable efforts. More laboratory 
work is evidently needed. 

In the Institute for Experimental Physics of Kiel University W. Lochte-Holt- 
greven and H. Maecker with their collaborators have developed a high current 
arc which, being stabilized by a whirl, burns very steadily and in which temperature 
and electron pressure can be measured quite accurately. By drastic cooling the arc 
can be contracted so that with currents up to 1200 A the temperature of its column 
rises to ~ 50,000°. 

Under conditions, which previous calculations by Vitense had indicated as 
favourable, Lochte-Holtgreven and his students (1951) found Wildt's H~ con- 
tinuum in the laboratory. G. Jurgens (1952) tested Holtsmark's theory for 
the broadening of the Balmer lines as well as Inglis and Teller's formula for the 
termination of the series by micro-fields under conditions of ^ 12,700°K and an 
electron pressure of ~ 0-15 atm, corresponding about to a white dwarf atmosphere ! 
It is hoped that similar light sources will allow measurements of transition proba- 
bilities for higher stages of ionization. Also the broadening of helium and other fines 
as well as the inducing of forbidden components by the electric fields of ions and 
possibly electrons offers a most promising field of research. 

One of the most disappointing problems in the theory of stellar spectra used to be 
the calculation of damping constants caused by collisions between the radiating atoms 
and neutral hydrogen atoms in later type atmospheres. 

Recently, however, progress became possible by suitably combining theoretical 
and experimental methods. Quantum mechanics indicate that the van der Waals 
energy of interaction &E between the radiating atom in a state k and the perturbing 
neutral hydrogen atom over a distance r can be calculated from the formula 



^E k 



hC, 



254 Fraunhofer lines and the structure of stellar atmospheres 

where the interaction constant C k is given approximately by 

C k = t . a^?. 

Here e, h have the usual meaning ; a is the polarizability of the perturbing atom, 
and B k 2 the matrix element of the (radius) 2 for the considered quantum state. 
Following Bates and Damgaard (1949) the B k 2 matrix elements can be calculated 
with fair accuracy even for complex spectra using the well-known hydrogen-like 
matrices in connection with the effective principal quantum number ri*. We have 



r* 2 

2Z 2 



** 8 = < • 7^i I 5 "* 2 + 1 ~ 3 ^ + !)}> 



where a means the Bohr radius, Z = 1, 2, ... for arc-, spark, . . . spectra, and 
1=0, 1,2, for s, p, d . . . states. 

The simplifying assumptions made in deriving the formula for AE k are essentially : 
(a) the energy diagram of the perturbing atom should have a large gap between the 
ground state and the excited levels near the ionization limit ; (6) the energy differ- 
ences between the level of the radiating atom and the nearer combining levels should 
be small compared with the ionization energy of the perturbing atom; and (c) the 
average over all possible relative orientations of the two colliding particles is formed 
classically. In general, these assumptions are fulfilled reasonably well. Weisskopf's 
quantum theory of collision damping further says that the broadening of a line should 
be determined by the difference of the interaction constants for its upper (a) and 
lower (b) level, i.e. G = G a — C 6 . It would be rather hopeless to investigate in the 
laboratory collision broadening of, say, Fe-lines by atomic hydrogen and to measure 
the corresponding collisional cross sections either for direct astrophysical application 
or for testing the described theory. However, if we substitute some rare gas, prefer- 
ably He, for H, we have only to change the well-known a's in the formula for the 
interaction energies AjE^ while everything else remains the same. So it appears 
possible to test the theory by measurements made with rare gases. One must only 
make sure that the applied pressure is low enough so that one is still dealing with two 
particle collisions, i.e. the ratio of the average distance r between two perturbing 
particles and the Lorentz collision radius p should be ^> 1. The table opposite 
gives a compilation of all the published measurements on line broadening by rare 
gases ; the figures for which the last-mentioned condition is well satisfied are printed 
in bold type. In general, the difference between measurements and theory keep 
within quite reasonable limits. More experimental work on line broadening, especi- 
ally in complex spectra and by helium under not too high pressures (for avoiding 
multiple collisions) would be highly desirable. 

In concluding this brief survey we should apologize for having left aside several 
aspects of our problem. We have not dealt with variable and peculiar stars and we 
have not spoken about magnetic fields and their importance for the dynamics of 
cosmical plasms. These fields, which in course of time will no doubt lead to important 
new vistas, will, however, be taken care of by more competent authors in this 
book. 



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L- L- 


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lO 


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U i~, 


Li L-, 


1 


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1 l_ 


ic- U, 








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02 


sp'sp' 1 


sp'sp ' 


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02 


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02 


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02 


02''o2" 


oa'aT 


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mm 


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in in 


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fefi PhS 



256 On the formation of condensations in a gaseous nebula 

References 

It is not intended to give here extensive references to literature. Instead we mention the following 

modern treatises: 

Astrophysics, edited by J. A. Hynbk (McGraw-Hill Book Co., 1951), especially Chapter 5, by B. 

Stromqren. 
Basic Methods in Transfer Problems, by V. Kourganoef (with collaboration of I. W. Busbridge) 

(Oxford University Press, 1952). 
Radiative Transfer, by S. Chandrasekhar (Oxford University Press, 1950). 
A revised edition of Physilc der Sternatmosphdren, by A. Unsold, with extensive literature references is 

in course of preparation (to be published in 1955). 



On the Formation of Condensations in a Gaseous Nebula 

H. Zanstra 

Sterrekundig Instituut, Universiteit van Amsterdam, Netherlands 
Summary 

In order to get an insight into the process of formation of condensations in an ionized gas of high kinetic 
temperature, two main principles are applied: 

( 1 ) that the pressure in the less dense medium 1 should be about the same as in the denser medium 2, 
which means that, in the denser medium, the kinetic or electron temperature is correspondingly 
lowered, and 

(2) that this lowering of electron temperature is brought about by electron impacts exciting low 
level lines. (If the optical depth of the condensations is large, so that the ionizing radiation is nearly 
completely absorbed, this effect may be enhanced.) 

The approximate balancing of pressure is verified in the case of prominences considered as condensations 
in the solar corona (Section 1). 

In order to account for condensations as observed by Baade in the planetary nebula NGC 7293 in 
Aquarius (Section 2) a very simplified model of a nebula is worked out, consisting of one substance only, 
occurring as ions A+ (for instance N++) and "atoms" A (for instance N+), the latter having a low meta- 
stable level, while the ionization of A into A+ is fairly complete. Introducing the liberation temperature 
T corresponding to the average energy of the photo -electrons when freed from A, it is shown that separa- 
tion into two phases 1 and 2 will occur above a certain critical liberation temperature T cr °f about 50,000°, 
the curves of constant 7* having a shape resembling those of the van der Waals theory. For T ^ 1 10,000°, 
or stellar temperature 170,000°, the volumes per unit mass in phases 1 and 2 have the ratio vjv 2 = 15, 
while the electron temperatures are T el = 130,000° and T e2 — 9000°, the latter being about the value 
derived in regions of planetary nebulae where forbidden lines are excited. Similar results can be expected 
for a nebula consisting of hydrogen with the substances A and A+ intermixed. 

It is suggested that the cometary shape of many of the condensations observed by Baade in NGC 7293 
might be due to aggregates of matter already present in interstellar space, serving as nuclei for condensa- 
tion, the tails being formed by dragging along of matter by the outward moving gases of the nebula. 



Introduction : The Mechanisms of Ionization and Recombination and of 
Electron Collision for a Gaseous Nebula 

The spectrum of many nebulae shows a number of bright lines, which have practic- 
ally all been identified as being produced by atoms and their ions, so that they can be 
described as luminous clouds of gas. Thus a planetary nebula consists of a rather 
regular mass of gas (disk, ring, etc.) at the centre of which a hot star is situated. Of 
this, the well-known ring nebula in Lyra is a typical example ; also the helical nebula 
in Aquarius, discussed in Section 3 of the present paper, belongs to this class. Accord- 
ing to present theories, the far ultraviolet radiation of the hot central star is absorbed 
by the gases in the surrounding envelope and ionizes the hydrogen atoms. These 
absorbing atoms are practically all in the ground state, since the nebula is much 



H. Zanstra 257 

larger than the star, so that the radiation incident on it is very diluted. By this , 
process the ultraviolet stellar radiation beyond v , the ionizing frequency of hydrogen, 
may be absorbed, and, for every absorbed quantum, one ionization occurs. Subse- 
quently, the photo-electron, thus freed, recombines on the various levels of the 
hydrogen atom and, in returning to the ground state, emits the various line spectra 
of hydrogen, among others the lines of the Balmer series, while captures on the second 
level produce the continuous spectrum at the head of this series. It is clear that, from 
the observed intensity of the hydrogen spectrum, one can estimate the number of 
ultraviolet quanta beyond v emitted by the star, and thus the stellar temperature, 
for which values of 20,000° and much higher are found (the star is approximated by 
a black body and the absorption of ionizing quanta assumed to be complete, though 
this may not be quite correct). This mechanism 1 of luminosity, ionization and 
recombination, applies to the spectra H, He i, He n, and others. 

Generally, the strongest lines in the nebular spectrum are however low level for- 
bidden lines, like the two green "nebulium lines" of the O in spectrum, or the two 
red lines of the N n spectrum. These cannot be sufficiently produced by mechanism 
1, but are believed to be excited by electron impact of the ion (0 ++ or N+ for the two 
examples) with the photo -electrons freed by mechanism 1. In other words, before 
recombining, the photo-electron may lose the major portion of their energy which 
passes into forbidden lines. This is stated as the principle of cooling in Section 2. 
This mechanism 2, excitation of low level lines by impact with photo-electrons, is there- 
fore a by-product of mechanism 1. It is also clear that the observed energy emitted in 
total by all forbidden lines is indirectly determined by the star's ultraviolet energy 
and gives another determination of stellar temperature. The fact that the stellar 
temperatures obtained by mechanisms 1 and 2 agree approximately furnished a 
check of the theory. 

Historically, the discussion of observations on diffuse nebulae (like the great nebula 
in Orion) and planetary nebulae by Hubble (1922) has been fundamental. His 
empirical rules induced the writer (Zanstra, 1926, 1927) to verify mechanism 1 for 
hydrogen. This mechanism was also conceived independently by Menzel (1926). 
After his very important identification of "nebulium lines" as forbidden lines of ions, 
the mechanism 2 was suggested by Bowen (1927, 1928) and checked quantitatively 
in the above manner by the writer (Zanstra, 1928, 1931a, 1931b) from his observa- 
tions of Victoria spectra of planetary nebulae and by Berman (1930) from similar 
observations of Lick spectrograms taken by him. 

1. Equality of Pressure at Different Electron Temperatures: 
Prominences in the Corona 

In 1946 the writer put forward and verified the idea that the pressure in a promi- 
nence of the quiescent type should be approximately the same as that of the sur- 
rounding solar corona (see Zanstra, 1947). Moving picture films of such prominences 
as were made by MacMath, Lyot, and the Harvard observers at Climax clearly show 
a predominantly downward motion of luminous material, which somehow must 
originate by condensation in the surrounding coronal material. This process of 
condensation is shown in a convincing manner in the coronal streamers (Pettit) 
which start from a point in the corona above the solar surface, and, equally convincing, 
in two-sided streamers, where the luminous material moves from a point above the 
solar surface in two opposite directions along an uninterrupted arc. This being the 

18 



258 On the formation of condensations in a gaseous nebula 

case, the pressure in the upper more extended portion of the prominence, where the 
spectrum for determining concentration and temperature is generally taken, should 
be balanced by the surrounding coronal material. 

If, in general, the less dense medium is indicated by the suffix 1, and the condensa- 
tion by 2, we have the requirement 

T n 

where T e represents the kinetic temperature or electron temperature and n the 
number of particles per cm 3 , which may be taken equal to 2n e , the concentration of 
electrons or protons. At an altitude of 50,000 km, the approximate height of a 
prominence, the observed values for the corona are about n el = 2 x 10 s , T el = 
7 x 10 5 and, for the prominence (Redman and Zanstra, 1952) n s2 = 2 x 10 10 , 
T e2 = 5 X 10 3 , so that ^2/^! = 100, T El /T e2 = 140, which shows that the requirement 
of equal pressure is approximately satisfied. 

In the next sections attention will be confined to condensations in planetary 
nebulae, where the same principle (1) will be applied. 

2. Condensations in the Planetary Nebula NGC 7293 (Baade). 
Cooling by Excitation of Low Level Lines 

Dr. Baade has observed a great number of very remarkable bright condensations in 
the planetary nebula NGC 7293, the helical nebula in Aquarius. They were mentioned 
by Dr. Menzel in a colloquium at the Zurich meeting of the International Astro- 
nomical Union in 1947 (Menzel, 1950). 

The temperature T* of the central star, the distance D and radius r n of this nebula 
are approximately (Zanstra, 1932) 

T* > 85,000°, D = 350 parsecs, r n = 105,000 astr. units (2) 

The stellar temperature is definitely a lower limit and may be a few times 100,000°. 
The photographs were taken with the 100-in. Mount Wilson reflector in the red 
region of Ha. and [N 11] AA6548, 6584 with exposures of f and 4 hours. We are much 
indebted to Dr. Baade for having placed prints at our disposal and for his comments. 
Fig. 1 shows an enlarged reproduction of such a red photograph, which was taken in 
1952 at the 200 in. Mount Palomar reflector by Dr. R. Minkowski. Like Dr. 
Baade's photographs it shows the ring, in reality three turns of a helix to be largely 
made up of condensations. In particular towards the inside of the ring there is a 
radial structure with respect to the central star, and at the inside blurred cometary 
shapes consisting of a head and a tail directed away from the central star are recog- 
nizable. In the nearly dark space inside the ring there are a fair number of small 
condensations with a typical cometary shape : a round head with a fairly long tail 
directed radially away from the central star. They do not occur in the neighbourhood 
of the central star. These "comets" in the dark space form the most startling feature 
of the photograph. They contribute however only a very minute fraction of the total 
brightness of the nebula on the photograph, which is practically given by the outer 
ring. Table 1 shows the emission lines in the photographic region observed by 
Minkowski (1942) for the brightest part of the ring north-east of the central star 
with intensity, excitation potential, ionization potential and the ionization potential 
of the next higher ion. In Dr. Baade's opinion it seems likely that in the red region 



H. Zan&tha 



259 




Fig. J. The planntary nnlmln X(JC l'2H',i 
(lied photograph by K. Minkiw^ki, Mount Wilson and Palomar Observatories.) 



260 



On the formation of condensations in a gaseous nebula 
Table 1. NOG 7293 



Line 


H /* 


H y 


Ha 


[0 m] 


[O m] 


[On] 


[Ne in] 


[Ne in] 


Red 
H a 


Red 
IN II] 


Red 

[Nil] 


X. 


4861 


4340 


4101 


5007 


4959 


3727 


3790 


3869 


6563 


6548 


6548 


Int. 


10 


41 


21 


60 


21 


31 


3 1 


8-4 


— 


— . 


— 


Ex. P. . 


12-07 


12-76 


13 05 


2-51 


2-51 


3-33 


3-20 


3-20 


10-20 


1-90 


1-90 


I.P. . 


13-60 


— 


— 


54-93 


— 


3515 


64 


— 


13-60 


29-60 


— • 


I.P.+ . 


— 


• — 


— ■ 


77-39 


— 


54-93 


972 


— ■ 


— 


47-43 


■ — ' 



in which his photographs were taken most of the light would originate from the two 
[N n] lines also given in the table. Since the intensity of Ha is generally about five 
times H/?, or fifty units, this statement indicates that the light of these [N n] lines 
together might well be in excess of the total light of the forbidden lines in the 
photographic region. 

Altogether this indicates that most of the light in the ring or the condensations 
(medium 2) originates from the forbidden lines of various kinds. These lines are 
all low level lines excited by the mechanism of collision by free electrons, and this 
means that the electron temperature in the condensations (medium 2) will be 
considerably lowered with respect to medium 1 : 



T A < T B 



• (3) 



and then equality of pressure (1) leads to n 2 > n x , which would explain qualitatively 
the formation of condensations viz. regions of high density. 

In the next section this principle of cooling by excitation of low level forbidden 
lines will be formulated quantitatively for a simple model and indeed will be shown 
to lead to the formation of two phases. 

3. Simplified Model of a Nebula with Condensations 

The ideas developed in the previous section will now be applied to a very simplified 
model of a planetary nebula. The treatment is quantitative, but the model is idealized 
and the assumptions approximate to such an extent that a direct quantitative com- 
parison with observations is hardly justified, the main purpose being to show the 
possibility of the formation of two phases in the nebular material, provided the 
temperature of the central star is high enough. 

The nebula is considered to consist of one substance only, say nitrogen, occurring 
only in two stages of ionization : the lower ions or "atoms" A and the next higher 
ions A+. The higher ions A+ = N ++ have no low levels, but A = N+ has a low 
metastable level B excitable by electron impact, giving [N n] lines. (In the case of 
oxygen, A+ = 0+++, A = 0++, emitting [0 in] lines.) 

The radiation incident on the element of the nebula considered has an intensity 
I v dv ergs per cm 2 per sec. within the frequency interval v to v-{-dv and is but incom- 
pletely absorbed by the nebular material under consideration, if the element is not 
too extended. 

Let n A be the number of "atoms" per cubic centimetre, and a„ the atomic absorp- 
tion coefficient for the frequency v. The number of ionizations per cubic centimetre 
per second is then 



N 



= "' J, *- i 



dv, 



(4) 



H. Zanstea 261 

where v is the ionizing frequency of A and v ' is either infinity or the ionizing fre- 
quency of the next state A + .* 

According to Hebb and Menzel (1940) the number of exciting collisions per cubic 
centimetre per second which raise the atoms A from the ground level to the 
metastable level B are 

*AB = ^*o»An±e-K ..-.(5) 

9 A X £ 

where J% = 8-54 x 10~ 6 a universal constant, Q. AB a factor dependent on the kind 
of atom and excited level of excitation energy X AB , g A the weight factor for the ground 
level, n e the number of electrons per cubic centimetre and T e the electron temperature. 

The average energy of the electrons which are present in 1 cm 3 is then ikT e , which 
might also have served as a definition of the electron temperature. Let, in a similar 
way, %kT represent the average energy of the photo-electrons when freed from the 
ground state of A, and f kT s the average energy of the free electrons wher^recombining 
on all levels of A+, then T will be called the liberation temperature and T e the capture 
temperature. (We may expect T to be of the same order as the temperature T* of the 
central star, and T s of the same order as T E , see formulae (15) and (16).) The average 
energy lost per electron between ionization and recombination is then lk(T — T £ ). 

The energy lost per cubic centimetre per second, disregarding loss by electron 
switches, all goes into the excitations of the forbidden line, so that 

mk(T-T e ) = ^ AB X AB , ....(6) 

provided de-exciting collisions are negligible. 

After substitution of (4) and (5) in (6), the electron concentration is expressed by 






k v- v ~dv 



X AB 

n e = — ^ — . Ti(T - T £ )e w.. .... (7) 

^AB cp y 

9a 
Let the "atom" A be an ion of charge i — 1, so that A+ has charge i. Then 
n e = in A+ + (1 — i)n A . It will be assumed that, for all concentrations considered, 
the ionization is nearly complete, so that approximately 



n c = in 



A+ 



> n A , n e -{- n A+ = yi + jj n £ . .... (8) 

Let m A be the mass of the neutral atom. The concentration p in grams per cubic 
centimetre, or the volume occupied by 1 g is then, using (7) and (8), given by 

x 4 



~ = P = m A n A+ 

where 



^», = ^Z.(i-)V* ( r-f,)e + .T, (9) 

i i \X AB / 



K = ^ — . ....(10) 

ki^*jr 
9a 

* For a planetary nebula, the incident radiation is short wavelength ultraviolet, supplied by the central star, but in general 
it may be ultraviolet radiation of any origin. For incident corpuscular radiation the results of the following derivation are 
equally valid, provided one replaces the integral in equation (4) by the number of ionizations it produces per cubic centi- 
metre per second on one atom A. Interstellar gas colliding with an expanding nebula may be considered as such corpuscular 
radiation, incident with the relative velocity (Oort, 1946). 



262 On the formation of condensations in a gaseous nebula 

The pressure in dynes per square centimetre p = (n e + n A+ )kT e similarly becomes 

p = (l + ]) nJcT. = (l + ]) X AB K (/-) V e 3 (f - T e )e + ^s (ll) 

Introducing the unit of temperature u T degrees Kelvin, the unit of mass u m grams 
and the unit of pressure u p dynes per cm 2 denned by 

Y 1 f 1 \ 

u T = ~~, u m = - m A K, u = I 1 + - X AB K, (12) 

K % \ 1/ 

with K given by equation (10), the equations (9) and (11) assume the simple form 

1 ■ 1 



v 



P = Ti(T-T E )e + Te, ....(13) 

P = ^ = T*(T - T e )e + T*, ....(14) 

where, according to the definitions, p represents the number of mass units u m per cm 3 
and v the volume in cm 3 occupied by the mass u m . 

For a given incident diluted stellar radiation, the average energy of the photo- 
electrons when freed is fixed, so that also the liberation temperature T = constant. 
Then, since T s must be a function of T e (see below), equations (13) and (14), by means 
of this variable parameter T e , express p as a function of v, which is represented in the 
p-v diagram as a curve of constant liberation temperature, but of course variable 
electron temperature. From considerations based on Kramers' absorption law 
Eddington (1926) has shown that, for stellar black body temperature T*, 

T* = |f, i£'kT < ionization energy, (15) 

and that in general 7* is a function of T such that T* > f T. 

Considering similarly Kramers' law for capture of electrons, one may show that 

T E = #f £ , if lcT e < ionization energy, (16) 

and that in general T s is a function of T £ such that T e > %T t . Now, in virtue of the 
cooling effect by the excitation of the forbidden line, T e and consequently T e 
decreases as the density increases or the volume decreases, and eliminating this 
variable parameter from equations (13) and (14) the equation of state p — p(v,T) is 
obtained. Using the approximation (16) T s = |T e , the equation of state becomes 

p = (T — %pv)p : 'v'e vv . (17) 

For actual computation p and v are best calculated from equations (13) and (14) for 
a set of T e values at a given T = f T* (16). 

In Figs. 2 to 4 the curves T — constant thus obtained for various values of T are 
plotted. The value of the electron temperature T e is also indicated for each point. 



H. Zanstra 



263 




Fig. 2. p-v diagrams for liberation temperatures T = 1, 1-986, 3, and 4-5. The volume per 
unit mass is v. The units of temperature, mass, and pressure (12) are used, which vary from 
curve to curve. The electron temperature T e varies along each curve and is indicated at 
various points by the corresponding figure. Above the critical liberation temperature 1-968 
separation in two phases 1 and 2, represented by the points A x and A 2 occurs 

At higher T (or stellar temperature T* > §T) each curve shows a maximum and a 
minimum, which coincide at a certain critical liberation temperature 

T er = 1-968, p cr = 3-53, v er = 0-310, at T s = 1-086, .... (18) 

as may be obtained analytically from equations (13) and (14) with (16). For lower 
values of f the maximum and minimum have disappeared. For a liberation tempera- 
ture above T cr the part between the two maxima is unstable, since dvjdp is positive : 
with increasing volume the pressure increases also and the expansion continues, 
similarly a decrease of volume results in a collapse by decreasing pressure. Stable 
equilibrium can only occur when the pressure is the same throughout the system, 
which means that only two phases, a dilute phase 1 and a denser phase 2, co-exist 
(see Section 1). The volumes per unit mass v x and v 2 of these two phases are found as 
the extreme points of intersection A x and A % of a horizontal line with the curve of 
constant T such that the areas cut off above and below this line are equal.* The 



* One cannot apply thermodynamical considerations. But, assuming continuity of the various states according to the con- 
tinuous curve of constant T when passing through the boundary region between the two phases, passage of the mass dm from 



■£• 



phase 2 to phase 1 through the boundary layer results in a mechanical energy freed dm I pdv. If in a hypothetical experiment 

te 

this is brought about by having the gas in a cylinder transparent to radiation and moving the piston outward, the work done 
on the piston is dmp{v x — v s ), where p is the equilibrium pressure. Since there is equilibrium, these two quantities should 
be equal, which results in the above equality of areas. If in this hypothetical experiment p is larger than its equilibrium 
value, which means that the horizontal line is drawn higher, one sees from the figure that the mechanical energy freed upon 
expansion is smaller than the work on the piston required to keep up this larger pressure, so that the pressure must go down 
upon expansion and thus come closer to the equilibrium value. Similarly one can show that also for too low initial p or 
compression one goes towards the equilibrium. Thus the equilibrium is always stable. However, the above assumption 
regarding the continuity of states requires further justification, since the variable pressure in the boundary layer could only be 
kept up by some field of force. 



264 



On the formation of condensations in a gaseous nebula 



IOO 



1 


i 




i 


■ — i — 


1 


1 


■ T 


" 










^s|0 










■ 










■vT-IO 








4* 




5/ 








^^^2 




. 


A 2 














"^Al 




' fc. 
















^^ 






h 














■ I 


















l\ 




r 














\ 
















' 


05 




i 




















i . 


> 








' 



OOS 



OIO 



OI5 



Fig. 3. The same as Fig. 2, but for T = 10 units 



700 



bOO 



500 



400 



300 



200 



IOO 
P 



- 




177 


1 

OO 


i — 


1 — 


... , , 




i — 


i /b 








\T-20 








- 






\o 






^^25 






- 


















- 


A 2 












a; 






i 5 












- 


{ 
















- 


\* 




> 


i. 




■ 


■ 


_ . i 


, 


0675 ) 



OC H 002 003 004 005 OOb 007 008 009 

Fig. 4. The same as Fig. 2, but for T = 20 units. Maximum and 
minimum are marked by crosses 



whole situation is analogous to the isotherms of the van der Waals Theory, except 
that the separation into phases occurs above a critical temperature of liberation and 
that there is no thermodynamical equilibrium ; in fact throughout there is dissipation 
of energy and, though the pressures in the two phases are equal, they are at different 
kinetic temperatures. 

Let v be the volume of one unit mass of mixture consisting of x units of phase 1 and 
1 — x units of phase 2. Then v = xv x + (1 — x)v 2 , so that one gets the well-known 
expression 

1 — x v x — v A X A 
x v — v 2 A 2 A' 

where A represents the point on A X A 2 corresponding to v. 



H. Zanstba 



265 



Table 2 contains, for a given T, the electron temperatures and ratio of the volumes 
per unit mass for the two phases. 

Table 2. Electron temperatures and volume ratios for the 
two phases 

The unit of temperature X AB /k is 21,800° for [Nil], A6572 
and 28,800° for [O in], 2.4991 (see Table 1). 













T . 


3 


4-5 


10 


20 


T sl . 


30 


51 


12-5 


26 




0-46 


0-35 


0-23 


018 


vjv 2 


6-5 


15 


50 


140 



The temperature unit is about 25,000°, being the average for the green [O in] and 
the red [N n] lines. Above the critical liberation temperature (18) of about 50,000° 
separation in two phases may occur. While T rises from 75,000° to 500,000°, the 
electron temperature T e2 of the condensations goes down from 11,500° to 4500°. This 
is not far from the electron temperatures around 10,000° derived by Menzel, Aller, 
and Hebb (1941) from the relative intensities of [0 in] lines in planetary nebulae. 
These authors in the case of the nebula in Draco NGC 6543 find a value even as low 
as 6000°. For the highest values T of 10 or 20, that is kT (in ordinary units) 10 or 20 
times the excitation energy 1 AB , the application of equation (5) may be somewhat 
doubtful. It is better founded on T= 4-5 = 110,000°, which corresponds to a 
stellar temperature of about 170,000°, a reasonable value for the central star of 
NGC 7293 (Section 2), though it might be higher. The theoretical electron 
temperatures according to the table are then about 130,000° for the dilute 
medium and 0-35 units or about 9000° for the condensations, which would be 
fifteen times as dense as the medium. Though not much stress should be laid 
on the actual values, the theory as developed in the foregoing seems to account 
for the formation of dense and very luminous condensations in NGC 7293, pro- 
vided that the concentration of matter is such that v in Fig. 2 is between 
A x and A 2 . 

For an actual nebula, the assumption of only one substance A, A+ is probably too 
idealized, since it also contains much hydrogen. Assuming hydrogen H, H+, mixed 
with the substance A, A + , equation (4) is replaced by 



N 



n 



+ n H \ 

Jv JvAH) 



.(4a) 



Jv„ n A jv i 



Jv,, 



the second integral referring to hydrogen. In equation (7), j is then to be replaced 

From considerations based on the Kramers theory, or, less 

generally, on the ionization equation for diluted black-body radiation, one may then 
show that, for not too high T s , the ratio of the two terms is independent of T £ , 
and this again leads to equations (13) and (14), however with definitions of the units 
u m and u v which dhTer from equation (12), provided that the ionization, both for 
hydrogen and the substance, is nearly complete. The curves of constant T given in 
Figs. 2 to 4 then remain valid. 



266 On the formation of condensations in a gaseous nebula 

4. Influence of Absorption of Ionizing Radiation. Remarks on the 
Tail Formation for the Comet ary Condensations in NGC 7293. Possible 

Future Applications 

According to the mechanism proposed in Section 3, separation in two phases can only 
occur if the smoothed out density p is such that v, or 1/p, is between the points A x 
and A 2 of Figs. 2 to 4, say the point halfway. But even then a condensation, p 2 = l/v 2 , 
can only be formed on a nucleus, the dimensions of which are large compared with the 
mean free path and also large enough for thermal conductivity to be inappreciable. 
One must therefore assume that, when the nebula first originates, there are such 
accidental regions of increased density of sufficient dimensions. Once such a nucleus 
is present, the condensation will grow in size until it gets so large that all ionizing 
radiation forming photo -electrons is practically exhausted by absorption. If the 
photo-electrons originate from hydrogen (with which some material A, A+ is inter- 
mixed) the condensation consists of nearly completely ionized hydrogen (H+ region) 
passing rather abruptly into an inner core where the hydrogen is practically neutral 
(neutral H region), and of very low electron temperature.* Assuming equality of 
pressure of the condensation (phase 2) and the core, the latter (region 3) would have a 
high density or small volume per unit mass, since v ± : v 2 : v z ~ T el : T s% : T eZ . In any 
case the formation of this neutral H core would mean that the condensation might 
progress a good deal further. 

In this way more or less globular condensations might be formed, so that this 
would account for the heads of the cometary structures observed by Baade in 
NGC 7293 (see Section 2). Near the inside of the ring his photograph shows these 
heads to have a diameter of about 1000 astronomical units, farther inward they are 
smaller. But the condensation mechanism in itself is not capable of explaining the 
tails, which have a length about 3000 astronomical units, nor can it account for the 
radial structure of the outer ring. Now Biermann ( 1951 ) has recently explained long 
straight tails of comets by the action of solar corpuscular radiation (known by 
magnetic storms) which imparts momentum to the gases in the heads. It seems that 
a similar explanation might apply to the tails of the cometary structures in NGC 7293, 
assuming an outward movement of the nebular gases, which consists of an expansion 
with a velocity increasing with the distance from the central star. Such an expansion 
for planetary nebulae was put forward by the writer (Zanstra, 1931c, 1932) on the 
basis of broadening and doubling of emission lines observed by Campbell and Moore 
and theoretical considerations, and it is now generally recognized to occur for many, if 
not all, planetary nebulae. If one assumes aggregates of matter to be present in inter- 
stellar space before the planetary nebula has developed to its present state, such aggre- 
gates might serve as nuclei for condensation as considered above, when the neb- 
ular gases reach them, and at the same time matter dragged along from them by these 
gases might form a streak on which the tail might further condense. Head and tail 
would further develop by condensation from the moving medium, carrying with it its 
momentum, so that finally the whole cometary structure would share the outward 
motion of the medium. The condensations should thus move radially outward with 



* These H+ and H regions are well-known from Stkomgren's treatment (1939) of interstellar hydrogen. A similar develop- 
ment for planetary nebulae had been given earlier by the writer (1932, pp. 142, 143). The ionizing radiation incident on the 
condensation originates primarily from the central star. But it should not be forgotten that in our case the nebula (medium 1) 
surrounds the condensation and that recombinations in this nebula on the ground level of the hydrogen atoms produce an 
emission of L e , the continuous spectrum beyond the head of the Lyman series, which amounts to about 40 per cent of the 
radiation absorbed by them. This stray L„ radiation, incident in all directions on the condensation, is added to the direct 
stellar radiation. 



H. Zanstba ^ 267 

a considerable velocity, perhaps of the order 50 km per sec, so that the suggestion 
might be submitted to an observational test. For this explanation the original 
aggregates would probably have to be small clouds of gas, or else consist of solid 
particles like ice crystals which easily can be converted into a gas, either by the stellar 
radiation or by the impact of the oncoming gases. For actually working out the 
problem, use could be made of the treatment by Miss Kltjyver (1951) of the collision 
of an expanding shell of gas with an interstellar cloud. 

At present the two main mechanisms for condensation of matter into a denser 
cloud, leading eventually perhaps to the formation of stars and planets, are gravity 
of the cloud and radiation pressure on the solid particles in it by stellar radiation from 
outside stars, the latter according to Whipple or Spitzer. The present treatment 
adds to this another mechanism: formation of two phases of different density and 
electron temperature in an originally homogeneous medium exposed to radiation, the 
gas pressure being operative in producing the equilibrium. 

Besides the application to planetary nebulae given above, one might think of 
possible applications to clouds formed in interstellar matter, where very low levels 
can be excited by electron collisions, as considered by Spitzer and Savedoff (1950). 
Spitzer (1951) has shown that the gas pressure of such clouds may be approximately 
the same as that of the surrounding interstellar medium of much higher kinetic 
temperature and much lower density. For this reason an interpretation in terms of 
two phases would be rather tempting, but the writer, who made some preliminary 
attempts, is rather doubtful whether this would be successful. 

Future applications to prominences might also be considered (see Section 1) but it 
seems probable that this problem is more involved. It is even conceivable that in 
this case the system is not exposed from outside to strongly ionizing radiation I v , but 
that material of high kinetic temperature is continually supplied to the corona and 
removed as the denser and colder phase in the prominences, so that there would be 
a stationary state of moving matter rather than an equilibrium. 

The writer is greatly indebted to Dr. W. Baade for having placed information 
regarding NGC 7293 at his disposal and for criticizing his views at an earlier stage, 
and who, in particular, stressed the idea that the condensations occurred in the 
nebular material itself rather than being there ready made before the nebula was 
formed. 



References 

Bebman, L 1930 Lick Obs. Bull., No. 430. 

Biebmann, L 1951 Z. Astrophys., 29, 274. 

Bowen, I. S 1927 Nature (London), 120, 473. 

1928 Ap. J., 67, 1. 

Eddington, A. S 1926 The International Constitution of the Stars, 

p. 377 (Cambridge). 

Hebb, M. B. and Menzel, D. H 1940 Ap. J., 92, 408. 

Hubble, E 1922 Ap. J., 56, 162, 400. 

Kltjyver, Helen A 1951 Chapter 12 of Problems of Gosmical Aero- 
dynamics (Proceedings of the sympo- 
sium on the motion of gaseous masses 
of cosmical dimensions, held at Paris on 
16th to 19th August, 1949) (Central 
Air Documents Office, Dayton 2, Ohio). 

Menzel, D. H 1926 Publ. Astron. Soc. Pacific, 38, 259. 

1950 Trans. Internat. Astron. Union, 7, 468. 

Menzel, D. H., Alleb, L. H. and Hebb, M. H. . 1941 Ap. J., 93, 230. 



268 The calculation of atomic transition probabilities 

Minkowski, R 1942 Ap. J., 95, 243. 

Oort, J. H 1946 M.N., 106, 159. 

Redman, R. O. and Zanstba, H 1952 Circular No. 6 of the Astron. Inst, of the 

University of Amsterdam, Table 3 
(from Proc. K. Ned. Akad. Wet., 
Ser. B, 55, 598). 

Spitzer, L. 1951 Chapter 3 of Problems of Cosmical 

Aerodynamics (see Reference Kluyver). 

Spitzer, L. and Savedoff, M. P 1950 Ap. J., Ill, 593. 

Stromgren, B 1939 Ap. J., 86, 526. 

Zanstra, H 1926 Phys. Rev., 27, 644. 

1927 Ap. J., 65, 50. 

1928 Nature (London), 121, 790. 

1931a Publ. Dominion Astrophys. Obs., Victoria, 

4, 209. 
1931b Z. Astrophys., 2, 1. 
1931c Z. Astrophys., 2, 329. 
1932 M.N., 93, 131, Table I. 
1947 Observatory, 67, 10. 



The Calculation of Atomic Transition Probabilities 

R. H. Garstang 

University of London Observatory, London 

Summary 

Recent calculations on the strengths of spectral lines are reviewed. Transitions permitted in Russell- 
Saunders coupling are discussed with particular reference to absolute strengths. Mention is made of 
studies of the effect of departures from Russell -Saunders coupling. Calculations on forbidden lines are 
outlined. Some of the difficulties of the subject are discussed. Appendices contain detailed references 
to recent work on transition probabilities and atomic wave functions. 



1. Introduction 

During the last twenty years immense progress has been made in the study and 
interpretation of astronomical spectra. The identification of forbidden lines in the 
spectra of the gaseous nebulae, solar corona, and upper atmosphere opened up a 
new phase of spectroscopic analysis. The development of high dispersion spectro- 
graphs and the increasing number of large telescopes available have led to the detailed 
analysis of the spectra of many individual celestial bodies. The development of the 
theory of stellar atmospheres, and especially the discovery and exploitation of the 
curve of growth technique have enabled quantitative information to be extracted 
from the mass of observational data which can now be obtained. In order to relate 
the observed intensities of spectral lines to the number of atoms responsible for their 
production a knowledge is required of the atomic parameters involved. In particular, 
reasonably accurate values of the atomic transition probabilities are needed for very 
many lines. Such data can be obtained in some cases by laboratory experiment, 
while in others recourse must be had to theoretical calculations based on quantum 
mechanics. In this paper we shall review the calculations which have so far been 
carried out. 



R. H. Gabstang 269 

2. Permitted Lines: Russell-Saunders Coupling 
The general theory of the calculation of the strengths of permitted lines has been 
fully described by Condon and Shortley (1952). Under the assumption of Russell- 
Saunders coupling it has been shown that the "strength" S of an electric dipole line 
is given by 

s = sr{M)sr(L)o\ ....(i) 

where Sf(M) is a numerical factor depending on the particular multiplet of a transi- 
tion array and Sf( L) is a numerical factor depending on the particular line of a multi- 
plet. £f{M) can be evaluated from two sets of tables computed by Goldberg (1935, 
1936) and Sf{L) from the tables of White and Eliason (1933) or of Russell (1936). 
The /-values and spontaneous transition probabilities A (sec. -1 ) associated with a 
line of wavelength X (Angstroms) and strength S are given by the formulae 

304£ 



A 



2-02 x 10 18 £ 
eo 2 A 3 



where a> 1 and co 2 are the statistical weights of the lower and upper levels of the line, 
respectively. 

If we adopt the central field approximation (in which we regard the total wave 
function of the atom as a suitable combination of one electron central field wave 
functions) then it can be shown that 

^s^rlW' ••■• (2) 

where I is the greater of the two azimuthal quantum numbers involved and - P if - P f 

r r 

are the radial parts of the wave functions of the jumping electron. The determination 

of line strengths in Russell-Saunders coupling is thus reduced to the calculation of a 2 . 

Many attempts have been made to determine a 2 . Among early work may be 
mentioned that of Trumpy and others based on the WKB method (Condon and 
Shortley, 1935, Chapter XIV). The most important calculations have been based 
on the self-consistent field procedure devised by Hartree and described by him in a 
comprehensive review (1947). An immense amount of labour is required to carry 
through calculations of this type and even when it is done the results are not always 
as satisfactory as might be desired. The principal weakness of the self-consistent 
field method is its neglect of correlation between electrons. A number of attempts to 
improve the approximation have been made by Biermann and Lubeck (1948, 1949) 
and by Biermann and Trefptz (1949a), who modified the central field by the intro- 
duction of a polarization potential proportional, for large r, to r -4 . The final energies 
obtained agree more closely with observation than those from self-consistent fields. 

Opportunity is taken to refer to some recent unpublished investigations by 
Douglas (1953). Using "EDSAC", the electronic calculating machine at the 
Cambridge University Mathematical Laboratory, he has developed methods of 
calculating atomic wave functions and has applied these to Si iv, an ion with a single 
electron outside a closed core. The novel feature of his work is the use of a polariza- 
tion potential given not analytically (as used by Biermann) but as a tabulated 



270 The calculation of atomic transition probabilities 

function. The use of EDSAC enabled over seventy trial functions to be investigated 
and a function was eventually obtained which reproduced the observed energies of 
six levels to within the accuracy set by the assumed field. This is very satisfactory as 
far as it goes. It is not yet clear, however, whether the potential used is uniquely deter- 
mined and much further work on other simple systems will be needed before the method 
can be extended to more complex cases. It will be interesting to examine transition 
probabilities calculated from his wave functions when these become available. 

Green (1949) and Green and Weber (1950) have made extensive calculations of 
/-values for Ca n deriving wave functions by mechanical integration of the wave 
equation on relay calculating machines. It is hoped that this work (and other 
similar investigations) can be extended to other complex systems and more use made 
of modern calculating machines as these become available. 

The most important of all recent work on the calculation of atomic transition 
probabilities is that of Bates and Damgaard (1949). They have shown that in 
calculating the transition integrals (a) it is permissible to neglect the departure of the 
potential of an ion from Coulomb form. This enables the transition integrals to be 
systematically tabulated in terms of (essentially) the energy parameters of the initial 
and final terms. For the simpler atomic systems (especially those with one electron 
outside a closed shell) the method gives remarkably accurate results, and it seems 
probable that the results are as good as or better than those obtained by the use of 
more elaborate methods referred to earlier. Lack of suitable comparison data makes 
it difficult to assess the accuracy of the method for more complex systems for which 
the Coulomb approximation would be at best of doubtful validity. Such information 
as is available, however, suggests that the method may be quite useful even for the 
heavier atoms. 

3. Permitted Lines: Departures from Russell-Saunders Coupling 

All the calculations on permitted lines so far described have been based on the 
assumption of Russell-Saunders coupling. Most of the atoms whose spectra are of 
astrophysical interest show substantial departures from Russell-Saunders coupling. 
It is desirable to investigate these departures which are usually discussed in two 
parts, intermediate coupling and configuration interaction. 

(a) Intermediate Coupling 

In intermediate coupling we include in the Hamiltonian terms arising from electron 
spin. It is normally sufficient to include only the spin-orbit interactions. The general 
theory of line strengths in intermediate coupling has been given by Condon and 
Shortley (1935). The first major attempt at a detailed calculation was made by 
Shortley (1935) for the 2p & Ss-2p 5 3p array in Ne i. The results were not completely 
satisfactory because the final strengths were sensitive to the adopted values of various 
parameters and these were not accurately known. Nevertheless the final results were 
in much better agreement with the available observational data than were values 
computed on the basis of Russell-Saunders coupling. 

Similar calculations for a complex spectrum of great astrophysical importance 
have been completed by Gottschalk (1948). He has calculated fine strengths for the 
3d 7 ( 4 P)4,9-3d 7 ( 4 P)42> and 3d 7 ( 4 P)4s-3d 7 ( 4 P)4p arrays in Fe I using intermediate 
coupling theory and neglecting configuration interaction and the interactions between 
terms based on different parents. Although some discrepancies remain, the agree- 



R. H. Garstang 271 

ment obtained between the calculated strengths and the available laboratory data is 
remarkably good. 

Much further work by these methods could and should be undertaken. Garstang 
has investigated the line strengths of the arrays 2p 2 3s-2p 2 3p in n, 2p*3s-2p*3p in 
Ne ii, 3pHs-3pHp in S n, 3pHs-3pHp in A n, 2p 2 3p-2p 2 3d in O n, and 2p*3p-2p*3d 
in Ne n. In the first three of these arrays the departures from Russell-Saunders 
coupling are quite small. In the other three arrays there are a number of large 
departures from Russell-Saunders coupling and many intercombination lines appear. 
Satisfactory strengths have been computed for many of these lines. 

It is important to note that all intermediate coupling calculations give relative 
line strengths normalized to the same total as those Russell-Saunders values from 
which they were derived. This is equivalent to saying that, in formula (1), a new 
value of S?{M)Sf{L) is found for each individual line, while Sy(i!/)y(i) for the 
whole array and the value of a 2 remain unaltered. 

(b) Configuration Interaction 

The effect of configuration interaction on line strengths has been investigated in a 
series of very important papers by Biermann and Trbfftz (1949b) and Trefftz 
(1950) for Mgi and by Trefftz (1951) for Ca I. These calculations, like those for 
intermediate coupling, have to be performed for one atom at a time and generaliza- 
tion is difficult. For the resonance lines of Mg i and Ca i for which earlier results 
calculated by self-consistent field methods (with exchange) are available, the general 
effect is to lower the /-values and transition probabilities by factors of the order of 
30 per cent. Much additional work is required on calculations of this type. Trees, 
Green, Boys, and others have made extensive studies of the effect of configuration 
interaction on the energy levels of various atoms. Detailed extensions of their work 
to transition probability computations will be awaited with much interest. 

4. Forbidden Lines 
It should be stated that in this connection the term "forbidden" is used as synony- 
mous with "multipole", the lines with which we are concerned arise from magnetic 
dipole or electric quadrupole radiation. We do not include intercombination lines 
which arise from departures from Russell-Saunders coupling but which are neverthe- 
less due to electric dipole radiation. 

The identification of many of the most important lines in the spectra of the gaseous 
nebulae provided a strong stimulus to study the theory of forbidden radiation. Many 
attempts were made to calculate the transition probabilities of these lines, none of 
the interesting lines being susceptible to laboratory production. The most important 
tabulation of numerical data was by Pasternack (1940), whose work superseded 
previous calculations. He studied the configurations p 2 , p 3 , p*, d 2 , and d 3 which 
produce most of the astronomically interesting lines. These lines are strictly forbidden 
for electric dipole radiation by the Laporte Rule (the transitions are allowed only 
between states of opposite parity). Some transitions are allowed for magnetic dipole 
and electric quadrupole radiation even in Russell-Saunders coupling and others 
appear in intermediate coupling. The detailed theory of these transitions was worked 
out by Shortley (1940) and numerical developments given by Shortley, Aller, 
Baker, and Menzel (1941) enable transition probabilities to be calculated for the 
p n configurations. The work of Pasternack remains the only source of data on the 



272 The calculation of atomic transition probabilities 

d 2 and d? configurations. Osterbrock (1951) has computed transition probabilities 
for a number of additional lines of astronomical interest. 

The calculations so far discussed were based on the theory of intermediate 
coupling taking as the perturbation the spin interaction of an electron with its own 
orbit. In general the results so obtained gave intensity ratios of various lines in good 
agreement with astronomical observations. 'In one important case, however, the 
theory failed to agree with observation. In O n the ratio of the intensities of the 
pair of nebular lines at 23727 due to the transitions l S- 2 D was widely different from 
the calculated intensity ratio. This discrepancy was explained by Aller, Ufeord, 
and Van Vleck (1949). In the p z configuration the lines 4 S- 2 D are forbidden for 
both magnetic dipole and electric quadrupole radiation to the first order in the spin- 
orbit interaction, but have non-vanishing intensities when the second order spin-orbit 
effects are included. Aller, Ufford, and Van Vleck showed that for this particular 
transition the effect of the spin-other-orbit and spin-spin interactions between pairs 
of electrons is large and comparable with the second order effect of spin-orbit inter- 
action. Only the latter effect had been included in the earlier calculations of Paster- 
nack and others. Other ions in the p 3 configuration which have been studied with 
special reference to spin-other-orbit and spin-spin effects include N I by Ufford 
and Gilmour (1950) and S n, n, and N I by Garstang (1952a). Garstang (1951b), 
and also Naqvi (1951), have examined the effects in thep 2 and^> 4 configurations of a 
number of ions. It has been shown that while the energy levels are appreciably 
affected there is little change in the transition probabilities. Similar calculations for 
other ions have been made by Obi (1951) and Garstang (1952b). 

These investigations seem to provide a satisfactory basis for the calculations of the 
strengths of forbidden lines. So far as the writer is aware no attempt has been made 
to examine what effect, if any, would be caused by the extension of the calculations 
to include the effects of configuration interaction. There do not appear to be any 
discrepancies between theory and observation which might be ascribed to configura- 
tion interaction, but the absolute values of the line strengths might be affected. The 
quadrupole lines are comparatively sensitive to the adopted quadrupole moments, 
and these in turn to the wave functions used. They would be more susceptible to 
perturbing interactions than magnetic dipole lines. It should be emphasized that all 
the transition probability calculations for forbidden lines of the magnetic dipole 
type give absolute values, subject to the uncertainties of certain parameters which 
are normally obtained by an empirical study of the energy levels. The electric 
quadrupole lines are subject to the same small uncertainties in addition to the very 
approximate quadrupole moment. Uncertainty of the quadrupole moment appears 
to be the most doubtful factor in the calculations of the strengths of forbidden lines. 
The most important task remaining is the computation of the strengths of some 
lines in more complicated configurations. Some work has been begun on the d* and 
d* configurations in Fe v and Fe in by Garstang but no results are yet available. 

5. General Discussion 
Reference must be made to two points of difficulty in the calculation of transition 
probabilities. The formula (2) for a 2 involves the dipole length integral 

^rP { P f dr. 



R. H. Gabstang 273 

It can be shown that this can be transformed into the dipole velocity form 

E Jo '"'Ir ,+ <W ' 

where E is the energy (in Rydbergs) of the transition and I is the larger azimuthal 
quantum number involved, and a third form (dipole acceleration) can also be obtained. 
These formulae would be exactly equivalent if P t , P f were exact solutions of the 
wave equation. When P t , P f are approximate the dipole velocity and dipole length 
integrals will have different numerical values. The importance of this difference was 
pointed out for H~ by Chandrasekhar (1945) and many subsequent investigators 
have calculated a 2 from the two formulae. It is not at all clear which value of a 2 
is the most reliable, or what mean value might be chosen. It is not even certain that, 
in general, the true value of a 2 lies between the two calculated values. The correct- 
ness of this last statement has been shown by Bates (1951) for the case of the 
lsa-2pa transition in H^. Exact wave functions for this molecule have been 
obtained and used to compute exact /-values (as a function of internuclear distance 
B). Approximate /-values have been obtained by applying the dipole length and 
dipole velocity formulae to wave functions obtained as a Linear Combination of 
Atomic Orbit als approximation. It was found that the dipole length formula gave 
greater and the dipole velocity smaller values than the true values. In many cases, 
unfortunately, the last conclusion appears to be uncertain or perhaps definitely 
false (Bates and Damgaard, 1949; Table 7). 

A second difficulty arises in several calculations. If, for example, we compute a 
self-consistent field for an atom we obtain as the appropriate eigenvalue of the 
Schrodinger equation a parameter which, for a system consisting of one electron 
outside a closed core, gives the best theoretical approximation to the energy of the 
corresponding spectroscopic level. This differs from the observed value. Should the 
theoretical or experimental energies be used in calculations of /-values ? If in a 
particular case, as for example the intermediate coupling calculations mentioned 
earlier, the labour involved prohibits obtaining theoretical values for the energies of 
the centres of gravity of the various terms, will the use of the observed energies give 
satisfactory transition probabilities, bearing in mind the well-known disagreement 
between the observed multiplet intervals and any calculations which neglect con- 
figuration interaction ? 

This problem has been investigated in an interesting paper by Green, Weber, 
and Krawitz (1951). They calculated oscillator strengths for all important transi- 
tions from the 3d state of Ca n using wave functions with and without exchange and 
using both calculated and observed energies. No experimental data was available 
for comparison, and recourse was had to the /-sum rule and partial /-sum rule. It 
was shown that the use of the calculated rather than the observed energies gave 
results in much closer agreement with the sum rules. This conclusion held whether 
wave functions with or without exchange were used, and whether dipole length, 
velocity, or acceleration formulae were used. The results obtained using observed 
energies were very poor for some transitions. 

It should be remembered that the /-sum rule is only an approximation based on 
the assumption that the series electron moves in a fixed field. The general agreement 
of the sum rule with the sum of the /-values calculated both with and without 
exchange and using calculated energies, together with the marked disagreement 



274 



The calculation of atomic transition probabilities 



between the results with and without exchange when observed energies are used, 
suggests that, at least for a one electron spectrum, the sum rules are valid with fair 
accuracy. It is important to note that while failure to satisfy the sum rules implies 
unreliability of the /-values, the satisfaction of the sum rule does not necessarily 
imply that the /-values are accurate. This is well shown by the results of Green, 
Weber, and Krawitz, whose individual /-values using calculated energies with and 
without exchange differ greatly although the /-sums are nearly equal. See Seaton 

(1951) for further discussion of /-sum rules. 

These last remarks illustrate what, in the writer's opinion, is the greatest difficulty 
in the whole subject of transition probabilities, whether theoretical or experimental. 
We refer to the almost complete absence of any transition probabilities of absolutely 
certain accuracy for any complex atoms. We have discussed some of the theoretical 
difficulties and uncertainties. Many of the experimental determinations, while of 
immense assistance to astrophysicists are nevertheless subject to serious uncertainties. 
The "absolute" determinations of Estabrook (1952) and Htjldt and Lagerquist 

(1952) for A4290 of Cr i, for example, differ by a factor of 100. This may be a parti- 
cularly bad case, but we cannot regard as satisfactory a situation in which such 
discrepancies exist. An experimental programme of especial interest is one being 
developed at Kiel. New high-temperature stabilized arcs have been constructed. 
Transition probabilities for carbon by Maecker (1953) are among the first results 
of this work. Much more work must be done to improve the accuracy of both 
theoretical and experimental determinations, to supply accurate experimental 
comparison data for checking theoretical work for atoms of moderate complexity 
(such as Ca i) and, above all, to supply numerical data which is greatly desired by all 
astronomical spectroscopists. 

Acknowledgment 
The writer would like to thank Mr. A. S. Douglas for communicating his results in 
advance of publication. 

APPENDIX I 



Atomic Transition Probability Data 
For a summary of the information, available in 1947 see Unsold (1938, pp. 128 and 
191) and Minnaert (1950). For recent work on forbidden lines see: Aller, 
Ufford, and Van Vleck (1949); Gold (1949); Ufford and Gilmour (1950); 
Obi (1951); Naqvi(1951); Osterbrock (1951); Garstang (1951b, 1952a, 1952b). 
Bates and Seaton (1949) list work on continuous absorption up to 1949. Bates 
and Damgaard (1949) give absolute line strengths for a number of atoms. We list 
below, with references, all other work on permitted transitions, theoretical and 
experimental, known to the writer which is not mentioned by these authors. 

Continuous absorption 



Ion 


References 


Ion 


References 


Ion 


References 


i 

Ion 


References 


Oi 
Fi 


49 
49 


Nei 
Nai 


49, 39 

50, 16 


Mg ii 
Sin 


8 
8 


Kn 
Can 


48 
31, 32, 50 



R. H. Garstang 
Line absorption 



275 



Ion 


References 


Ion 


References 


Ion 


References 


Ion 


References 


H 


64 


Mgi 


9, 10, 56 


Sill 


1 


Mn i 


36 


Li i 


55 


Mg ii 


1, 7, 


CI ii 


1 


Fei 


12, 29, 38 


Be i 


9 


All 


7 


An 


1, 25 


Fe xiv 


26 


Ci 


39a 


Aim 


1 


Ki 


7, 54, 61 


Nil 


18, 37 


Cn 


1, 7, 39a 


Sin 


7 


Cai 


57 


Rbi 


54 


Nn 


1 


Sim 


1 


Can 


31, 32, 50, 58 


Bai 


62 


On 


1, 25 


Si iv 


1, 17, 33 


Vi 


46 


Hgi 


11, 41 


Ne ii 


20, 25 


Pra 


1 


Cn 


18, 19, 35, 36 


Tli 


54 


Nai 


7, 50, 54 


Sn 


1, 25 











APPENDIX II 

Calculation of Wave Functions 
A bibliography of wave functions has been given by Hartree (1947). 
short supplementary list of wave functions published since 1947. 



We append a 



Ion 


References 


Ion 


References 


Ion 


References 


Ion 


References 


Cn 


7 


Mgll 


7 


Sin 


7, 8 


Can 


57 


Nn 


50a 


All 


/ 


Ki 


7, 41, 61 


Ca xm 


15 


Ne in 


21 


Aim 


36a 


Km 


48 


Fe xiv 


26 


Nai 


7 


Alrv 


36a 


Cai 


57 


Hgi 


41 


Mgi 


9, 10, 56 















Unpublished wave functions are available from A. S. Douglas (of the Cambridge 
University Mathematical Laboratory) for S n and Si iv, from D. R. Hartree 
(Cavendish Laboratory, Cambridge) for Ne I computed by B. H. Worsley and from 
the present writer for Ne n. 



[20 
[21. 



References 

Aller, L. H 1949 

Aller, L. H., Ufford, C. W. and Van 

Vleck, J. H 1949 

Bates, D. R 1946 

Bates, D. R. and Damgaard, A 1949 

Bates, D. R. and Seaton, M. J 1949 

Bates, D. R. . . 1951 

Biermann, L. and Lttbeck, K 1948 

Biermann, L. and Lttbeck, K 1949 

Biermann, L. and Trefftz, E 1949a 

Biermann, L. and Trefftz, E 1949b 

Brossel, J. and Bitter, F 1952 

Carter, W. W 1949 

Chandrasekhar, S 1945 

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276 



The calculation of atomic transition probabilities 



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M. F 

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1952 Ap. J., 115, 206. 



Atomic Collision Processes in Astrophysics 

H. S. W. Massey 

Department of Physics, University College, London 



1. Introduction 

The development of modern astrophysics owes a great deal to the great extension of 
knowledge of atomic phenomena. Although the great majority of the applications 
of atomic physics in this direction have been concerned with radiative processes of 
one kind or another there are a number of circumstances in which a knowledge of 
the rates of collision processes involving electrons, ions and/or neutral atoms and 
molecules is required. 

One very important example of the need for accurate knowledge of collision cross 
sections associated with electron impact is provided by the theory of the solar corona. 
Apart from its own intrinsic interest a knowledge of the quality and quantity of the 
radiation emitted from the outer layers of the solar atmosphere is necessary for 
the interpretation of solar-terrestrial relations. The identification by Edlen (1942) 
of the coronal lines as due to the 2 P 4 — 2 Ps transition in the ground configuration of 
Fe xiv showed that the kinetic temperature of the corona must be of the order 
10 6 °K. Further evidence for this remarkable result has since been provided from 
observation of the radio-noise spectrum emitted from the quiet sun (Martyn, 1946). 
An atmosphere at this high kinetic temperature must emit radiation of very short 
wavelength, probably in the soft X-ray region. Hoyle and Bates (1948) have shown 
that a suitable emission of such radiation could well produce terrestrial atmospheric 
ionization similar to that observed as the F layer. Radiation from deeper layers of 
the solar atmosphere, such as the chromosphere, may also be important in producing 
ionization in the earth's atmosphere in excess of that expected from the usual black 
body model of the Sun as radiator, and at a different altitude. To investigate the 
possibilities theoretically it is necessary to have available reliable information on 
cross sections, not only for radiative recombination of electrons to highly ionized 
atoms, but also for ionization and excitation of such atoms by electron impact. This 
may be seen by reference to one of the pioneer investigations of the solar corona, 
by Woolley and Allen (1948). 

They consider first the nature of the equilibrium in the corona where the kinetic 
and radiation temperatures differ by several orders of magnitude. The ionization 
equilibrium must arise from a balance between ionization by electron impact and 
radiative recombination. If there are xn atoms per cubic centimetre in the (s + l)th 
and n{ 1 — x) in the sth stage of ionization the equilibrium equation is 

nxn e <x = n(l — x)n e y, 

where a is the coefficient of radiative recombination of electrons to the (s + l)-fold 
ionized atom and y that for further ionization (by electron impact), n e being the 
number of electrons per cubic centimetre. This gives 

x y 

1 — x a 

277 



278 Atomic collision processes in astrophysics 

Hence if y and a are known as functions of electron energy it is possible to determine 
under what conditions x approaches unity for a particular ion. In the solar corona 
this must be so for Fe xiv and this means that, from y and a, the kinetic temperature 
of the corona can be determined. With this information available it is possible in 
principle, from a knowledge of the appropriate excitation cross sections and transition 
probabilities, to calculate the intensity of the various characteristic radiations emitted 
from coronal atoms in terms of their concentrations and that of the electrons. 

A programme of this kind was carried out by Woollby and Allen (1948) using 
the best available information about the collision cross sections. They obtained 
many results of importance ; the coronal kinetic temperature comes out to be close 
to 10 6 °C, evidence that the main contribution to the coronal line emission in the 
ultraviolet comes from the principal lines of Mg x was obtained, and the intensity of 



Fig. 1. Energy level diagram (O m) 



< 


4363 


5007 


1 


4959 

3 


■ 


' 





\ 



*p 



the continuous emission was estimated to be less than the number of Lyman quanta 
emitted by the chromosphere. It is therefore important to verify that the assump- 
tions made about the atomic constants are valid, including particularly the ionization 
and excitation cross sections. 

The development of a theory of the chromospheric emission offers much greater 
difficulties. A start has been made by Giovanelli (1949) and by Woolley and 
Allen (1950). The former author investigated the hydrogen spectrum of the sun 
by attempting an analysis of the population of the different atomic hydrogen levels 
in the chromosphere. It is impossible to carry out this analysis without knowledge of 
the cross sections for production of various transitions between the low-lying levels 
of the hydrogen atom by electron impact. Whereas in the coronal problem the 
transitions are optically allowed, in the chromospheric case optically disallowed 
transitions such as ls-2s are important. In addition, at the kinetic temperature of 
the chromosphere, the exciting electrons have energies only slightly in excess of the 
threshold for the transition concerned. The distinctive feature of such collisions is 
that electron exchange effects are often of dominating importance whereas in 
optically allowed transitions this does not apply. 

An accurate knowledge of the population distribution among the low-lying 
hydrogen states is also important in connection with the possibility of observing the 
2 2 s 1 — 2 2 2?., transition as a line emission in solar radio-noise at 9882 Mc per sec 
(Wild, 1952). The effectiveness of collisions in determining the distribution of atoms 
between the 2s x , 2p± and 2p § states is the decisive factor in this case. 

The effectiveness of electron impact in producing transitions from the ground state 
to the metastable states of atoms is also a major factor in the determination of the 
intensity of emission of many principal lines in nebular spectra. The classical 
example is provided by the "auroral" and nebular lines of O in. Fig. 1 illustrates 



H. S. W. Massey 279 

the levels arising from the ground configuration of this ion. The green nebular lines 
X 5007 A and X 4959 A arise from the W 2 — 3 P 2 and X D 2 — 3 P X transitions and the 
auroral line A 4363 A from X S Q — X D 2 . Under nebular conditions effective electron 
collisions are sufficiently frequent to ensure that the population of the three levels 
of the ground 3 P term follows a Boltzmann distribution about the electron tempera- 
ture (Hebb and Menzel, 1940). The two metastable levels ^ and W 2 are populated 
by electron excitation from the 3 P states. Deactivation occurs by radiation and by 
superelastic collisions. It follows that the relative intensity of the auroral and 
nebular lines is determined by the relevant transition probabilities and collision cross 
sections and by the electron temperature (Menzel and Aller, 1941). The absolute 
intensity of either fine depends further on the concentration of O in 3 P ions and of 
electrons in the nebula. The electron concentration may be obtained from the 
intensity of the Balmer continuum emitted by the nebula. 

Hence, if the radiative transition probabilities and collision cross sections are 
known from atomic theory it follows that measurement of the absolute intensities 
of the nebular and auroral O in lines from a nebula enables one to determine the 
O ni abundance in that nebula. If this procedure could be applied to all states of 
ionization the atomic constitution of the nebula could be obtained. Even though this 
is not possible for all states of ionization it can be carried out for many atomic ions 
and the results extrapolated in a reasonable way. The pioneer work in this field is 
that of Menzel and his co-workers (1941, 1945). We shall see, however, that some 
of the collision cross sections they employed in the analysis of the O ni forbidden 
line intensities were too large. 

The green and red forbidden lines of O i which arise from similar transitions to 
those of in (see Fig. 1 (b)) are a prominent feature of airglow and auroral spectra. 
In the auroral case electron impact undoubtedly plays an important part in deter- 
mining the relative intensities at different heights in the earth's atmosphere but in 
the airglow other processes of excitation are responsible for populating the 8 and 
D states. 

Although, in general, collisions between systems which are both of atomic dimen- 
sions are not of such importance under astrophysical conditions as are electron 
impacts, there are phenomena in which such collisions play a vital role. The light 
emitted and the ionization produced by meteors in the earth's atmosphere are due to 
inelastic impacts between atoms evaporated from the meteor and atmospheric 
atoms and molecules. Again, protons are more effective than electrons in the 
chromosphere in producing transitions between the 2s and 2p levels of hydrogen 
(Purcell, 1952). 

The recent observations of Gartlein (1951) and by Meinel (1950) of Doppler- 
shifted hydrogen lines in auroral spectra have confirmed the expectation that 
energetic protons are among the primary particles whose entry into the earth's 
atmosphere gives rise to auroral displays. This makes it important to study the 
various processes, including particularly charge exchange, which can occur as the 
incoming protons penetrate into the atmosphere. Charge transfer on impact between 
ions and neutral molecules may also profoundly modify the ionic constitution of the 
ionosphere and it may be important in a similar way in the radiating regions of 
comets. Other types of collisions between atomic systems are also very important 
in the ionosphere. 

It is clear therefore that the development of atomic collision theory is important 



280 Atomic collision processes in astrophysics 

in many applications to astrophysical (including geophysical) phenomena. We shall 
now examine briefly how far this development has proceeded paying special attention 
to electron collisions for which the greatest progress has been made. 

2. The Theory of Inelastic Collisions op Electrons with Atoms 

There is a profound difference between the state of the theory of energy transfer on 
collision between material particles and that of radiative transition probabilities. 
In the latter case the only difficulty is that of determining with sufficient accuracy 
the wave functions which describe the initial and final states ; the formula expressing 
the probability in terms of these wave functions is quite reliable. On the other hand, 
when the electron energy is close to the threshold for the inelastic collisions concerned, 
the usual formulae for the collision cross section are quite unreliable and cannot 
be used with confidence even for collisions with atomic hydrogen for which the wave 
functions are known exactly. 

The usual, perturbation, formula which is employed in the first instance to calcu- 
late collision cross sections is that of Born. This is derived on the assumption that 
the motion of the impinging electron relative to the atomic system can be described 
by plane waves, both before and after the collision. A modification of this approxi- 
mation due to Oppenheimer allows for the possibility that electron exchange may 
take place during the collision but still employs the plane wave approximation. 
We shall refer to these two as the Born and OB approximations respectively. 

A detailed examination of the validity of these Born and OB approximations by 
comparison with experimental data has been made by Bates, Fundaminsky, 
Leech and Massey (1950) (B.F.L.M.). They also employed, as a further check, a 
general conservation theorem which states that the maximum possible contribution 
to the cross section for any inelastic collision from encounters in which the relative 
angular momentum is {1(1 + l)} 1/2 7& is P(2Z + l)/47r 2 , where A is the wavelength of the 
relative motion. It is seen from this analysis that a distinction must be made between 
inelastic collisions involving optically allowed and optically disallowed transitions. 
In the former case the Born approximation does not appear to give results which 
are grossly in error at any electron energy. At low energies it overestimates the cross 
section to an increasing extent but usually by not more than a factor of 2 (B.F.L.M., 
1950). The observed behaviour may be represented empirically by expressions of the 
form adopted by Woolley and Allen (1948) in their study of the solar corona and 
it is unlikely that any very serious error was introduced in their analysis from their 
assumptions in this regard. Exchange seems to be relatively unimportant even at 
very low energies. It is fortunate that the simplest approximations work fairly well 
for these cases as it is very difficult to improve them without very great labour. 

The situation is very different for optically disallowed transitions, particularly 
those in which there is no change in azimuthal quantum number of the active atomic 
electron. Inelastic or superelastic electron collisions of this kind are often among the 
most important in applications as they include the excitation or deactivation of 
metastable states. For collisions of this kind, electron exchange effects are usually of 
dominating importance and it is found that the OB approximation, the simplest 
that can be used, is usually wholly unreliable (B.F.L.M., 1950). Whereas in optically 
allowed transitions contributions come from so many relative angular momenta that 
the conservation theorem cannot be employed as a check, for the encounters we are 
now considering the major contribution, near the threshold where the cross section 



H. S. W. Massey 



■281 



is large, comes from impacts with a definite and known relative angular momentum. 
The conservation theorem may therefore be employed as a convenient and reliable 
check. It was found in this way that many of the earlier calculations carried out by 
the OB method were invalid, giving cross sections in excess of the limit X 2 (2l -j- l)/47r 2 . 
This included those used by Menzel and Aller (1941) in their determination of the 
abundance of O in in nebulae. Unsatisfactory results were also obtained when the 
test was applied to the excitation of the 2 S S level of helium. It has recently been 
found that the OB method also gives invalid results for the excitation of the meta- 
stable level of hydrogen (Erskine and Massey, 1952). 

Although the simple approximations are so very unsatisfactory for these cases 
there is much more scope for introducing improved methods. This is largely because 
most of the contributions come from one relative angular momentum. The most 
important modification which can be made is that of replacing the plane wave 
representation of the colliding electron by a wave distorted by the mean atomic 
field in which the electron moves. In determining this distortion exchange effects 
may or may not be included. If they are so included we refer to the approximation 
as the D.W.B.O. (distorted wave Born-Oppenheimer), if not, as the D.W.B. The 
calculation of the appropriate distorted waves has been greatly facilitated by the 
introduction of variational methods (Erskine and Massey, 1952 ; Httlthen, 1944). 

Both of these approximations still assume that the coupling between the initial 
and final atomic states is not strong. To investigate this Massey and Mohr have 
discussed a schematic model for which exact solutions may be obtained even if the 
coupling is strong, and compared with the predictions of the corresponding D.W.B.O. 
and B.O. approximations. It is found that, whereas the B.O. approximation is 
always unreliable, no matter how weak the coupling, the D.W.B.O. gives good results 
even when the coupling is so strong that the cross section rises to as much as one-half 
the limiting value. Their calculations show, however, that the distorted wave 
approximation is only reliable when the distortion is accurately allowed for. In 
short it will usually be necessary to employ the D.W.B.O. approximation in which 
exchange effects in producing distortion are included, and not content oneself merely 
with the D.W.B. 

Detailed calculations using the D.W.B.O. approximation have been carried out 
by Erskine and Massey (1952) for the excitation of the 28 level of hydrogen. Their 
results profoundly modify the predictions of the OB approximation. This may be 
seen by reference to Table I. 

Table I. Comparison of cross sections for 1S—2S excitation of 
hydrogen calculated by the OB and D.W.B.O. methods 



Electron energy 


Cross section in units wa 2 


Limit set by Con- 


(e-volts) 


OB 


D.W.B.O. 


servation Theorem 


10-2 
13-5 
19-4 
30-4 
54-0 



1-590 
0-503 
0104 
0-020 



0-178 
0-094 
0035 
0-011 


1-000 
0-694 
0-444 
0-250 



Note. — The cross sections given are those for excitation by electrons 
with zero angular momentum only. The contribution from higher 
angular momenta is very small at energies below 20 e-volts, but is impor- 
tant at higher energies. It can be calculated adequately by the OB 
approximation. 



282 Atomic collision processes in astrophysics 

Detailed analysis of the data shows that in this case the D.W.B.O. results, while 
not violating the conservation theorem and representing a great improvement in 
the OB results, may still be somewhat in error near the maximum. In other words, 
the coupling between the 1$ and 2S levels may be too strong. Some evidence 
supporting this is forthcoming from some preliminary calculations by Massey and 
Moiseiwitsch (I), in which a variational procedure is employed but no assumption 
made about the strength of the coupling. It is hoped to carry out a still more accurate 
calculation in which the coupled integro-differential equations concerned are solved 
numerically. 

The D.W.B.O. approximation is also being applied in detail, by Massey and 
Moiseiwitsch (II), to the excitation of the 2 3 S and 2 1 8 levels of helium. Preliminary 
results for the 2 3 S case are encouraging, whereas the OB method gives a maximum 
cross section near the threshold of l-57ra 2 , the D.W.B.O. reduces this to 0-047ra 2 
which is quite close to that derived from the rather imperfect experimental data. 

Even for much more complicated atoms such as oxygen it has been possible to 
obtain greatly improved results. Seaton has been able to calculate cross sections for 
the excitation of metastable states of p 2 and p* configurations by a method which is 
virtually an improved version of the D.W.B.O., applicable to transitions within a 
configuration, which allows for close coupling. In all cases the conservation theorem 
is no longer violated and the cross sections can be used with some confidence for the 
study of nebular abundances and auroral and airglow phenomena. 

It seems probable therefore that reasonably reliable data on electron collision 
cross sections will soon be available for many astrophysical applications. An 
experimental research programme is also being initiated at University College, 
London, to provide additional reliable observational checks on the theory. 

3. Collisions Involving Atomic Systems (Massey and Burhop, 1952) 
The situation is much less satisfactory when both colliding systems are of atomic 
dimensions. If the velocity of relative motion is high compared with that of the 
internal motion of the electrons in the atomic systems which are concerned in the 
transition there is no difficulty in principle as Born's approximation may be safely 
applied. Even then the computation of the integrals which appear in charge exchange 
cases is a tedious and lengthy task. When the opposite conditions apply, the velocity 
of relative motion being small compared with that of the atomic electrons con- 
cerned or, in other words, the time of collision is long compared with the effective 
period of the electronic motions, the problem is much more difficult. Under these 
conditions the probability of the particular inelastic collision concerned is small, the 
process being nearly adiabatic, but it is impossible to say at present from theoretical 
arguments how small it is for a collision with any particular relative velocity. Unfor- 
tunately in many cases of practical importance, as for meteor ionization, the collisions 
are of this nearly adiabatic type. Progress in predicting cross sections for processes 
of this kind must necessarily be slow. An extensive research programme devoted to 
this end is now in progress both at University College, London, and Queen's University, 
Belfast. The theoretical work, which is being carried out at both establishments, 
includes a detailed study of the excitation of hydrogen atoms by slow protons in 
which the perturbation producing the energy transfer is the kinetic energy of relative 
motion. The experimental work, being carried out in London, includes a systematic 
study of charge exchange between positive ions and neutral atoms and molecules and 



E. E. Salpeter 



283 



of electron detachment from negative ions on impact with rare gas atoms. This will 
be extended to study ionization by impact of ions and of neutral atoms. It is hoped 
that systematic research on these lines will eventually make possible the prediction 
of the rates of nearly adiabatic processes of sufficient accuracy for application. 



References 



Aller, L. H. and Menzel, D. H. . . 1945 
Bates, D. R., Fundaminsky, A., Leech, 

J. W. and Massey, H. S. W 1950 

Edlen, B 1942 

Erskine, G. and Massey, H. S. W. . . . 1952 

Gartlein, C. W 1951 

Giovanelli, R 1949 

Hebb, M. H. and Menzel, D. H. . . . 1940 

Hoyle, F. and Bates, D. R 1948 

Hue/then, L 1944 

Martyn, D. F. 1946 

Massey, H. S. W. and Burhop, E. H. S. . 1952 

Massey, H. S. W. and Mohr, C. B. O. . . 1952 

Massey, H. S. W. and Moiseiwttsch, B. L. 1953 

Meinel, A. B 1950 

Menzel, D. H. and Aller, L. H. . . 1941 

Pttrcell, E. M 1952 

Seaton, M. J 1953 

Wild, J. P 1952 

Woolley, R. v.d. R. and Allen, C. W. . 1948 

1950 



Ap. J., 102, 239. 

Phil. Trans. A., 243, 93. 
Ark. Mat. Astr. Fys., 28, No. 1b. 
Proc. Roy. Soc, 212, 521. 
Phys. Rev., 81, 463. 
M.N., 109, 298. 
Ap. J., 92, 408. 

Terr. Mag. and Atmos. Elect., 53, 51. 
K. fysiogn. Sollsk. Lund. Fork, 14, No. 21. 
Nature (London), 158, 632. 

Electronic and Ionic Phenomena (Oxford Univer- 
sity Press), Chaps. VII and VIII. 
Proc. Phys. Soc. (London) A., 64, 545. 

I. Proc. Phys. Soc. (London) A., 56, 406. 

II. Work in progress. 
Phys. Rev., 80, 1096. 
Ap. J., 94, 30. 

Ap. J., 116, 467. 

Phil. Trans. A., 245, 469. 

Ap. J., 115, 206. 

M.N„ 108 292. 

M.N., 110, 358. 



Energy Production in Stars 

E. E. Salpeter 

The Australian National University, Canberra, Australia* 
Summary 

Recent work and trends in the field of stellar energy production and related topics are reviewed. Thermo- 
nuclear reactions are discussed in general, and a list is given of those reactions which might be important 
in the interior of different types of stars, together with the temperature at which these reactions set in. 
It is shown that the proton -proton chain almost certainly is the source of energy of the Sun and of cooler 
main sequence stars, and the carbon -nitrogen cycle most probably the energy source for the hotter main 
sequence stars (A, B, O). Some current ideas on the evolution of stars, particularly of subgiants in 
globular clusters, are discussed. 



1. Introduction 

Let us consider first of all our best-known star, the Sun. We know accurately the 
amount of energy the Sun loses in the form of electromagnetic radiation per second. 
The rate of energy loss per unit mass of the Sun is not very large, about 2 ergs per gm 
per sec, modest even compared with the rate of heat production during animal 
metabolism. On the other hand, geological evidence tells us that the Sun has been 
radiating at approximately its present rate for an extremely long time. This rules 



* At present at Australian National University, on leave of absence from Cornell University, Ithaca, New York. 



284 Energy production in stars 

out chemical sources of energy, which could have supported the Sun's radiation for 
not much longer than the age of the Egyptian Empire. The energy released in the 
gradual gravitational contraction of the Sun is much more impressive and could have 
served as the Sun's fuel for tens of millions of years. Such a life-span seemed adequate 
not too many decades ago. But nowadays geologists and astronomers agree that the 
Earth and Sun are several thousand million years old and the only source of energy 
potent enough for such a long time is the energy released in nuclear transformations. 

During the 1930's Atkinson and Houtermans, Gamow and Teller, Bethe and 
Critchfield, and v. Weizsacker laid the foundation for the study of nuclear 
reactions in stars. They pointed out how the thermal energy of nuclei in the interior 
of stars took the place of man-made accelerators to initiate nuclear reactions and 
discussed which species of nuclei might be involved. In 1939 Bethe wrote a classic 
paper, analysing in great detail all the possible reactions involving fairly light 
nuclei. He proved that in stars not too different from the Sun only two reaction 
chains are of importance, the carbon-nitrogen cycle and the proton-proton chain. 
On the nuclear side the biggest advance since 1939 has been a vast increase in the 
experimental information on nuclear resonance levels and on cross-sections of 
nuclear reactions. Thus the numerical values for the rates of many relevant reactions 
are different and much more accurate than in Bethe's paper. But Bethe's main 
conclusions about the reactions transforming hydrogen into helium and supplying 
the star's energy for most of its lifetime still hold. At least this is the case for main 
sequence stars, which consist mainly of hydrogen and are in a steady state. On the 
astrophysical side, too, our quantitative knowledge of the chemical composition of 
main sequence stars and notably of the opacity in their interior has improved greatly, 
but without altering Bethe's picture qualitatively. One new trend, however, has 
been an increased interest in stellar evolution and hence in stars which do not lie on 
the main sequence. Of particular interest are very young stars which are still under 
gravitational contraction and very old stars which have exhausted a considerable 
fraction of their hydrogen supply. 

In Section 2 we give a brief review of the principles and results of calculations on 
the rates of various thermonuclear reactions of importance in different types of stars. 
In Section 3 we take stock of the present position of the study of the energy sources 
and interior structure of main sequence stars. In Section 4 we discuss briefly, in 
a highly tentative and oversimplified way, some current ideas on the evolution of 
stars. More detailed reviews of earlier work on some of the topics treated here will 
be found in: Bethe, 1938; Bethe and Marshak, 1939; Gamow and Critchfield, 
1949; Johnson, 1950; Chandrasekhar, 1951; Salpeter, 1953; Aller, 1954. 
The bibliography mainly contains recent papers which give references to earlier work. 

2. Thermonuclear Reactions 

We know that all atomic nuclei consist of neutrons and protons (nucleons), bound 
together by the very strong nuclear forces. We give below a table of binding energy 
(B) per nucleon in mega-electron-volts for a few typical nuclei. 

H 1 He 3 He 4 B 11 O 16 Fe U 

B 2-8 7-1 6-9 8-0 8-5 7-5 



E. E. Salpeter 



285 



One feature of such a table is that the binding energy B tends to increase with 
atomic weight until atomic weight 40-80 (Fe-region) is reached. Thus energy will be 
released, in general, in a reaction building up a heavier nucleus out of two lighter 
ones (up to the Fe-region). Another important feature is the fact that the He 4 -nucleus 
(alpha-particle) has a large binding energy compared to neighbouring nuclei. Con- 
sidering this fact and the fact that hydrogen is the most abundant element in the 
Universe, it is not surprising that main sequence stars live solely on the conversion 
of hydrogen into helium. 

All the reactions of stellar interest involve collisions between positively charged 
nuclei, close enough for the short range nuclear forces to come into play. At these 



80 



-2 









}c^ 








£-~7 


& 






\ 


,*& 

































5 IO 15 20 , 25 

TEMPERATURE C IN KD<> °IO 

Fig. 1 . The rate of energy production e (in ergs per gramme per second) as a function of 
temperature for the proton-proton chain and for the carbon-nitrogen cycles. The curves 
are plotted for (p# H 2 ) = 100 and (px s x CN ) = 1» where p is the density in grammes per 
cubic centimetre and x a , a; CN are the abundances (by mass) of hydrogen and carbon plus 

nitrogen, respectively 

very small separations the two nuclei experience a strong electrostatic repulsion, 
corresponding to a Coulomb barrier of about a million electron volts (MeV). At 
temperatures of the order of 10 7 °K the mean thermal energy is only about a thousand 
electron-volts. Most of the nuclear reactions actually occurring, involve nuclei 
whose kinetic energy lies in the "high velocity tail" of the Maxwell-Boltzmann 
distribution, but is still small compared with the Coulomb barrier energy. The main 
feature of thermonuclear reactions then is that the reaction rate involves two 
exponential factors, one a Maxwell distribution factor depending on temperature, 
the other a Coulomb barrier penetration factor depending on the nuclear charge. 
Consequently the reaction rate increases strongly with temperature and decreases 
strongly with increasing atomic charge (Bethe, 1938; Bethe and Maeshak, 1939). 
Further, the temperature dependence is stronger for higher atomic charge. 

As we mentioned before there are two chains of reactions, each having the same 
net result, namely,* 

4H 1 -> He 4 + 2v + 26 MeV. (1) 

The carbon-nitrogen cycle (Bethe, 1938) involves successive reactions between 
protons and the isotopes of carbon and nitrogen, which merely act as "catalysts" 
without being used up. The cross-sections for all the reactions involved in this cycle 
have now been measured in the laboratory, but all for kinetic energies larger than 
occur in stellar interiors. In Fig. 1 the present value for the rate of energy production 
on this cycle is plotted as a function of temperature (Fowler, 1954). These values 



* v stands for neutrino. 



286 Energy production in stars 

are probably correct to within a factor of two, but there is still a small possibility 
that one particular nuclear resonance level has been missed in the laboratory 
experiments, in which case the rate would be about twenty times larger than given 
in Fig. 1. 

The proton-proton chain involves the collision between two protons, accompanied 
by a beta-decay resulting in a deuteron (Bbthe and Critchfield, 1938). Such a 
beta-decay is very improbable, but, on the other hand, the penetration of the Coulomb 
barrier is much easier for singly charged protons than for the reactions of the C-N 
cycle. This p-p reaction cannot be observed in the laboratory but its rate of energy 
production can be calculated from the known properties of the two-nucleon system 
and from beta-decay theory (Salpeter, 1952a; Freeman and Motz, 1953). This 
rate, accurate to within about 20 per cent is also plotted in Fig. 1 . Note that at low 
temperatures the p-p chain predominates, at high temperatures the C-N cycle. 

At temperatures appreciably below 10 7 °K the conversion of hydrogen into 
helium proceeds at a very slow rate. But some other light nuclei, in the presence of 
hydrogen, react appreciably even at these lower temperatures. A deuteron, in 
particular, will absorb a proton and release energy in the form of a gamma ray 
already at about 10 6 °K. The various isotopes of Li, Be, and B absorb protons at 
slightly higher temperatures, an alpha particle being emitted in each process. The 
net result is the destruction of Li, Be, and B and the building up of He 3 and He 4 . 
The He 3 produced is in turn converted into He 4 . Many of the relevant reactions have 
already been investigated to some extent in the laboratory and more thorough 
investigations are under way. It is hoped that in the near future all these reaction 
rates will be known, at least to within a factor of two. 

He 4 cannot react with a proton, since Li 5 does not exist. Furthermore two He 4 - 
nuclei cannot react with each other, since Be 8 is unstable. The only process by means 
of which He 4 nuclei can be destroyed, is one requiring very high temperature and 
density (Opik, 1951a; Salpeter, 1952b), namely 

3He 4 -> C 12 + y + 7-3 MeV. . . . . (2) 

The detailed mechanism of this reaction is rather complicated, proceeding via 
intermediate steps involving the unstable Be 8 and a resonance level in C 12 which was 
only discovered quite recently. Once C 12 has been produced, it can absorb another 
He 4 -nucleus giving O 16 and a gamma ray. O 16 in turn can be converted into Ne 20 , 
Ne 20 into Mg 24 , etc. At present the rates of these reactions can only be calculated 
very crudely (any of the results could be in error by a factor of more than 100). 
But present indications are that C 12 , O 16 , and Ne 20 are produced in roughly comparable 
proportions, whereas the production of Mg 24 , etc., is much too slow. 

At much higher temperatures still even C, O, and Ne nuclei can react with each 
other, even in the absence of hydrogen and helium (Salpeter, 1952b ; Hoyle, 1946). 
A large number of different reactions can take place, including for instance the 
collision between two carbon nuclei. Most important are reaction chains in which 
some very energetic photons (from the "high energy" tail of the Planck distribution 
of black-body radiation) break up some nuclei which have a relatively low binding 
energy. In most of these processes alpha-particles or protons are produced which are 
again absorbed by other nuclei. The end result of such reaction chains is that less 
stable nuclei are destroyed and more stable ones built up. There is thus tendency 



E. E. Salpkter 



287 



to build up the nuclei which have the largest binding energy per particle, namely 
those in the iron region. 

In Table I we summarize some of the groups of reactions we have discussed. The 
temperatures stated are roughly the ones at which these reactions proceed at an 
appreciable rate. It should be noted that, although a large number of reactions are 
involved in the groups following the formation of helium, the total energy release in 
all these reactions is small compared with that of the H-He conversion. 



Table I. Required temperature and energy release of typical reactions 




Initial nuclei .... 


D 2 


He 3 , Li, Be, B 


H 1 


He 4 


C 12 , O 16 , Ne 20 


Final nuclei ..... 


He 3 


He 4 


He 4 


C 12 , O 16 , Ne 20 


Fe -region 


Energy release per nucleon (in MeV) 


3 


2 


7 


1 


0-5 


Temperature (in 10 s °K) 


1 


5 


8 


100 


(1 to 2) x 10 3 



3. Main Sequence Stars 

Let us assume we know the mass if of a star, the law of energy production in its 
interior (as a function of temperature, density, and chemical composition), and that 
the star is in complete equilibrium (temperature, radius, etc., steady). We can then 
calculate all the remaining physical properties of the star, including its radius R, 
absolute luminosity L, and central temperature T c , provided that we assume a 
specific "stellar model" and chemical composition. For a main sequence star the 
model involves the assumptions that most of the energy is produced in a small region 
pear the centre of the star and that the chemical composition is roughly uniform 
throughout the star. The chemical composition of stellar atmospheres is known at 
least roughly, the concentration (by mass) Y of helium being 5-30 per cent, the 
concentration Z of all heavier elements about |-3 per cent and the bulk of the star 
consisting of hydrogen. 

For the Sun such calculations have been carried out by several authors (Epstein 
and Motz, 1953; Naur and Stromgren, 1954), using the known solar mass M Q 
and the law of energy production for the proton-proton chain (see Fig. 1). The 
calculated values for R and L agree surprisingly well with the accurately known 
observational values. These results certainly confirm one's faith in the theory of 
stellar interiors and in the p-p chain as the energy source. In fact one can now invert 
the above argument, assume the observed values of M, R, and L and calculate the 
composition parameters Y and Z. The latest calculation for the Sun gives (Naur 
and Stromgren, 1954): 

T c = 13-5 x 10 6 °K, Y = 0-25, Z= 0-01. (3) 

The calculated values for Y and Z are, unfortunately, very sensitive to any small 
errors made in the computation. Thus the values in, for example, (3) are not yet 
very accurate, but it is hoped that reliable values will be available in the not-too- 
distant future — when accurate formulae for the opacity coefficients become available 
for numerical solutions on electronic computing machines. 

For other main sequence stars the picture is not quite as complete as for the Sun, 
largely due to the lower precision of the observational data. Some calculations 
have been done for red dwarf stars whose physical properties are fairly well known. 



288 , Energy production in stars 

Earlier calculations (Aller etal., 1952), using the accepted rate of energy production 
for the p-p chain, gave a slight but systematic discrepancy with observation, the 
calculated luminosity being higher than the observed one. Recent theoretical work 
on the effect of convective envelopes of red dwarfs (Osterbrock, 1953), however, 
predicts lower central temperatures (and hence lower calculated luminosity) than a 
simple calculation. According to this work theory and observation agree also for red 
dwarfs, to the accuracy expected. 

As will be seen from Fig. 1, the p-p chain dominates the energy production at low 
temperatures, the C-N cycle at high temperatures. For the Sun (Naur and Strom- 
gren, 1954), the p-p chain exceeds the carbon cycle's energy production by a factor 
of about 100, and for the cooler red dwarfs by an even larger factor. But for the 
hotter and more luminous main sequence stars (all O and B and most A stars) the 
carbon cycle should predominate. Only a few quantitative calculations have been 
done so far in this region of the main sequence. The physical properties of Sirius A 
can be fitted by assuming the rate of energy production given by the carbon cycle 
(Swihart, 1953). Qualitatively, at least, observations on even more luminous main 
sequence stars are also compatible with the carbon cycle. 

4. Stellar Evolution and Discussion 

There is at present no generally-accepted theory of the evolution of stars. However, 
the following crude picture, based mainly on oversimplified theoretical considerations, 
may at least serve as the basis of comparison with observation. 

In regions containing gas and dust under "suitable" conditions, condensations of 
various masses form. Such a condensation (initially still with fairly low density and 
temperature) begins to radiate and to contract gravitationally, its temperature, 
temperature gradient, and luminosity rising. The time taken from such a proto-star 
to shrink to its dimensions on the main sequence depends on its gravitational energy 
content and its luminosity throughout the contraction. Presumably a lower limit 
to this time is obtained by assuming a luminosity throughout the contraction equal 
to the main sequence value. This time is about 2-5 x 10 7 years for a proto-star of 
solar mass M Q , about 10 6 years for 3M G , etc. If the proto-star contained an appre- 
ciable amount of D, Li, Be, or B it will burn these elements at different stages of its 
contraction, increasing the total contraction time slightly. 

Once the star reaches its main sequence position, the energy production from the 
H -> He conversion keeps pace with the radiative energy loss and the contraction 
stops. The time taken for the conversion of only 1 per cent of the stellar mass from 
hydrogen into helium is as long as 10 9 years for a star of mass M Q , a few times 
10 7 years for 3M Q . 

Let us assume the star is poorly mixed, which seems likely at least for a considerable 
fraction of stars (Opik, 1951b). Hydrogen is then converted into helium almost 
exclusively in the hottest central regions of the star. Thus a core of different com- 
position (and mean molecular weight) from the rest of the star is formed. When this 
core contains about 10 per cent of the mass of the star, its configuration ceases to be 
stable (Schonberg and Chandrasekhar, 1942). Its core will contract and heat up, 
while its envelope expands and cools (Sand age and Schwarzschild, 1952). At 
least in the early stages of this development the star will become much redder and 
slightly more luminous (red giant or subgiant). Details of this development are not 
yet known, but it seems likely that in the centre of at least some such stars high 



E. E. Salpeter 289 

enough temperatures and densities are reached for the conversion of He into C, 0, 
Ne (and possibly even for the further reactions outlined in Section 2). It should be 
remembered that even after leaving the main sequence the bulk of the star still 
consists of hydrogen and the bulk of the energy production still comes from the 
H-He conversion in a thin shell just outside the core. But it is rather likely that 
the onset of further nuclear energy production in the centre of the core can alter the 
structure of a star. Such processes may be involved in some types of variable stars 
or supergiants. 

It also seems likely that at some late stage in the development of a star, it will 
become unstable enough to lose mass from its outer layers, either gradually or 
catastrophically. This loss of mass will presumably stop when the mass of the 
remaining core is less than the Chandrasekhar limit and this core will eventually 
become a white dwarf. A white dwarf contains essentially no hydrogen in its interior 
(Mestel, 1952), has too low a temperature for nuclear reactions involving helium 
or heavier nuclei, and gravitational contraction is prevented by the high pressure of 
the degenerate electron gas. Having no possible sources of energy left, the white 
dwarf will cool down very slowly and, like old soldiers and some famous generals, 
merely fade away. 

The above outline of stellar evolution at best represents a sort of average picture, 
since the evolution of different stars may well depend strongly on various factors. 
In particular, the structure of a star (as well as the timescale of the evolution) 
depends strongly on its mass and its development will be affected by its original 
rotational velocity, its magnetic field and by the presence or absence of accretion 
from nearby gas clouds (Hoyle and Lyttleton, 1949). But the structure of main 
sequence stars is well understood, the proton-proton chain certain as energy source 
for the Sun and cooler main sequence stars, the carbon cycle very probable for hotter 
ones. Much theoretical work has been done on the structure of red giants by 
Schwarzschild and his co-workers, Hoyle, Lyttleton, H. and C. M. Bondi, and 
others (for references see a recent paper by Roy (1952)). There probably are different 
types of red giant stars and the theory of Schwarzschild and Sandage outlined 
above may only apply to subgiants in globular clusters (and other systems belonging 
to the type II population). The absence of very luminous main sequence stars and 
the luminosity and colour of subgiants present in globular clusters (Arp, Baum, 
Sandage; 1952) agree well with this theory. 



References 

Aller, L., et al 1952 Ap. J., 115, 328. 

Aller, L. H 1954 Astrophysics, Vol. 2 (Ronald Press, New 

York). 

Arp, W., Battm, W. A. and Sandage, A. R. . . 1952 Astron. J., 57, 4. 

Bethe, H. A 1938 Phys. Rev., 54, 248. 

Bethe, H. A. and Critchfield, C. L 1938 Phys. Rev., 54, 248. 

Bethe, H. A. and Marshak, R. E 1939 Rep. Progr. Phys., 6, 1. 

Chandrasekhar, S 1951 Chap. 14 in Astrophysics, A Topical 

Symposium (A. Hynek, Ed., McGraw- 
Hill, New York). 

Epstein, I. and Motz, L 1953 Ap. J., 117, 311. 

Fowler, W. A 1954 Rev. mod. Phys. (in press). 

Frieman, E. and Motz, L 1953 Phys. Rev., 89, 648. 

Gamow, G. and Critchfield, C. L 1949 Theory of Atomic Nucleus, p. 264 (The 

Clarendon Press, Oxford). 



290- The effective pressure exerted by a gas in turbulent motion 

Hoyle, F 1946 M.N., 106, 343. 

Hoyle, F. and Lyttleton, R. A 1949 M.N., 109, 631. 

Johnson, M 1950 Astronomy of Stellar Energy (Faber, 

London). 

Mestel, L 1952 M.N., 112, 583 and 598. 

Naur, P. and Stromgren, B. 1954 Ap. J., 119, 365. 

Opik, E. J 1951a Proc. Roy. Irish Acad., A54, 49. 

1951b M.N., 111, 278. 

Osterbrock, D. E 1953 Ap. J., 118, 529. 

Roy, A. E 1952 M.N., 112, 484. 

Salpeter, E. E 1952a Phys. Rev., 88, 547. 

1952b Ap. J., 115, 326. 

1953 Ann. Rev. Nucl. Sci., 2, 41. 

Sandage, A. R. and Schwarzschild, M. . . . 1952 Ap. J., 116, 463. 

Schonberg, M. and Chandrasekhar, S. . 1942 Ap. J., 96, 161. 

Swihart, T. L 1953 Ap. J., 118, 577. 



The Effective Pressure Exerted by a Gas in Turbulent Motion 

G. K. Batchelor 

Trinity College, Cambridge 

Summary 

This paper examines the way in which the effective mean pressure exerted by turbulent motion in a gas 
varies when the gas is subjected to an adiabatic homogeneous distortion. The results depend on the 
directional properties of the distortion and of the turbulence, and on the speed with which the distortion 
is carried out. In certain simple cases the effective normal pressure due to the turbulence is proportional 
to the nth power of the mean density, where n has a value which sometimes is as small as unity 
and sometimes as large as three. The dependence of total pressure (thermal plus turbulent) on mean 
density may therefore change when the turbulent pressure becomes greater than the thermal pressure, 
i.e. when the turbulent velocity fluctuations become supersonic. 



1. Introduction 

A significant feature of current research in theoretical astrophysics is the emphasis 
laid on the existence of turbulent motion in large-scale gaseous bodies. This very 
proper emphasis is based on the notion that non-turbulent relative motion of a fluid 
is almost always unstable and that the energy of the random turbulent. motion usually 
grows until it is an appreciable fraction of the total kinetic energy. The general 
effect of the turbulence is to act as an exchange mechanism between different parts 
of the gas and to produce fluctuations in the values of macroscopic quantities — that 
is, to produce all those effects also produced by the random thermal motion of the 
discrete particles of the gas. The magnitude of the effect of turbulent motion of the 
gas will sometimes be larger than that of thermal motion of the particles, depending 
on the nature of the effect concerned. The intention in this note is to consider briefly 
the effective pressure exerted by the gas as a result of its turbulent motion, and to 
show that in certain circumstances turbulent fluid behaves under adiabatic com- 
pression as a polytropic gas. The corresponding index in the pressure-density relation 
seems in some cases to be larger than both the index for the thermal motion of the 
particles and the effective index for radiation, so that in such cases variations of the 
state of the gas when the density is large will be dominated by the turbulent pressure. 



G. K. Batchelor 291 

We begin with a demonstration, in the manner of Reynolds' classical work on 
turbulence, that the effective turbulent pressure is directly related to the kinetic 
energy of the turbulent motion in unit volume of the gas. Let v t be the i-component 
of the macroscopic, or continuum, velocity at any point in the gas, and p and p the 
gas density and pressure. Then the force equation for a small volume element of the 
gas acted on by an external force with components X { per unit mass is 

d P v t d P v i v j dp 

(repeated suffixes being summed) on the assumption that viscous stresses are 
negligible. If the motion is turbulent, we can write 

*\ = U i + u t , 
where U i is the average value of v t and u t is the turbulent fluctuation in velocity. 
(The averages involved in turbulent motion are best thought of as probability aver- 
ages, i.e. as averages over the values which the quantity would take if the whole 
dynamical development of the system occurred a large number of times ; occasionally 
averages over a large volume of fluid or over a large time will give the same results 
and will be more useful.) Then on taking the average of all terms in (1) we find 

IT + 1^- = pX < - *, {pd " + W'> + £ ' • • ■ • <2 » 

dpU t d 

8 = " IT ~ as {U ' pw ' + u » m ' ) ' 



where the term 



will usually be small compared with the other terms since the mechanical connection 
between fluctuations in density and velocity is weak. If e is neglected, equation (2) 
shows that the effect of the turbulence on the distribution of mean velocity is to add 
a term pu i u j to the stress tensor. It is in this sense that the turbulence adds to the 
effective "pressure" exerted by the gas; we note, however, that the thermal motion 
of the discrete particles of the gas is without appreciable directional preference so that 
the diagonal components of the thermal stress tensor are dominant, whereas the 
turbulent motion may not be isotropic and the non-diagonal components of pu^Tj 
may be important. 

This result is probably familiar to everyone. Let us now go on to ask two questions 
which are of interest in astrophysics. First, when is the turbulent pressure more 
important than the thermal pressure ? ; and second, how does the turbulent pressure 
change when the gas is expanded or compressed ? The answer to the first question 
emerges from the relation 

pU t U t p UM; 

t— 1 - 1 GH '—-t—t = yM 2 , (3) 

p p 

(y = ratio of specific heats of the gas), which shows that the two pressures have 
comparable magnitudes when the Mach number of the turbulence M (= ratio of 
root-mean-square velocity fluctuation to the velocity of sound corresponding to the 
mean density p and mean gas pressure p) is of order unity. Thus, as expected, the 
turbulent pressure is larger than the gas pressure when the turbulence is supersonic, 
i.e. when the turbulent velocity fluctuations are larger than the thermal velocities of 
the gas particles. 



292 The effective pressure exerted by a gas in turbulent motion 

The answer to the more important second question is not as evident. Two com- 
plicating factors occur in the problem. First, it is to be expected that the turbulence 
will be changed by compression of the gas in a way which depends on how the com- 
pression is carried out, so that an imposed strain of general type (although possibly 
still spatially homogeneous) must be considered in place of an isotropic expansion or 
compression. Second, the relaxation time of turbulent motion, i.e. the time required 
for turbulence to adjust itself to a new mean velocity distribution or to new boundary 
conditions, is not negligibly small as it is in the case of thermal motion of the gas 
particles, so that some account must be taken of the speed with which the distortion 
or strain is carried out. With regard to this latter difficulty, it does not seem to be 
possible to consider the general case, but the two extremes of very rapid and very 
slow homogeneous distortions have certain simple features. When the distortion 
takes place so quickly that the simultaneous relative displacement of elements of the 
gas produced by the initial velocity distribution (or produced by the external force 
acting on the new arrangement of the fluid) is small compared with the relative 
displacement produced by the distortion, all elements of the fluid are subjected to the 
same distortion since the gas behaves approximately as though it were initially at 
rest. The change in the velocity distribution (both mean and fluctuating) is then 
due to the consequent rearrangement of the vorticity, or angular velocity, distribu- 
tion. On the other hand, when the distortion is very slow, there will be sufficient time 
for complete adjustment of the turbulence to the new conditions presented at each 
stage of the distortion. We proceed to consider these two extreme cases separately. 

2. Cases of Rapid Distortion 
The simplest kind of distortion is a homogeneous isotropic strain, i.e. a uniform 
expansion or compression, in which all line elements of the fluid are multiplied by a 
factor e, say. The direction of the angular velocity vector at each point of the fluid 
is unchanged, and the requirement that the angular momentum of each portion of 
the gas shall remain unchanged implies that the velocity of each element of the gas 
is divided by the factor e. Hence if suffix denotes the initial state of the gas, 



= e _5 (p)o(^%)o 

t*i l^-l (pu^, ( 4 ) 

w 

showing that all components of the turbulent stress tensor behave, for adiabatic 
changes, like the thermal stress in a monatomic gas. This is not surprising, since 
there are no "internal" degrees of freedom of the fluid continuum assumed in dis- 
cussions of turbulent motion, and the energy of compression is distributed among the 
translational degrees of freedom in proportion to the energy they already possess. 

It is possible to work out the change in the turbulent stress tensor consequent 
upon a rapid distortion which is spatially homogeneous but which is not the same in 
all directions (Ribner and Tucker, 1952; Batchelor and Proudman, 1953). 
However, the details of the general case are rather complicated, and will not be 
reproduced here since the turbulent pressure does not vary with density in the simple 
manner of a polytropic gas. The reason for the complexity is that in general the 
direction of the angular velocity vector at any point in the fluid is changed by the 



G. K. Batchelor 293 

distortion, so that the changes in the three components of the velocity are not 
independent and cannot be considered separately. Two further special cases, addi- 
tional to the one leading to (4), in which the direction of the angular velocity vector 
at each point is unchanged, are (a) one principal extension ratio large compared 
with the other two, and (b) two principal extension ratios equal and large compared 
with the third extension ratio. We have to consider advanced stages of these dis- 
tortions, at which the angular velocity vector has already been turned considerably 
and has approximately attained its asymptotic direction. With further increase in 
the relative magnitude of the one principal extension in case (a) or of the two ratios 
in case (6), conservation of the angular momentum of any portion of the gas requires 
the velocity at any point to change according to the relation 

u { s« (Wf)o/ e » (no summation), .... (5) 

where the initial instant specified by suffix must now be an instant at an advanced 
stage of the distortion and e t is a (principal) extension ratio in the i-direction. Hence 

(W 3 ) 2/3/ ^ 5/3 



/5\ 5 / 3 

- I ( W**)o> (6) 

\Pn/ 



so that the relation between turbulent pressure and mean gas density can be a power 
law with an index either greater or less than 5/3, according to the choice of direction 
of the normal stress or pressure force. In the interesting case of a disk-shaped mass of 
fluid for which e x and c 2 are equal and large compared with e 3 , the normal stress in 
the direction of the axis of the disk is pu 3 2 , and 

/ei\ 4/3 /p\ 5/3 

P u z * & -I -I (pu iUj ) - 
\e 3 / VV 

Thus if the disk is flattened still further by decrease of e 3 with e x and e 2 remaining 
constant, we shall havep oc e z ~^, and the turbulent pressure offered to the boundary 
at right angles to the direction of the compression is proportional to the cube of the 
mean gas density. This large value of the index is made possible by the fact that 
the velocity components u x and u % in the plane of the disk receive a share of the energy 
provided by compression of the gas only inasmuch as the mean density increases. 

3. Cases of Slow Distortion 
At the other extreme, we can consider a distortion which is slow enough to permit 
complete adjustment of the turbulence to the new mean velocity distribution or 
boundary conditions presented at each stage of the distortion.* This requires 
specification of the source of the energy of the turbulence (there must be a continual 
replenishment of the turbulence energy for the case of a slow distortion to have any 
significance, since unreplenished turbulence would simply dissipate itself and dis- 
appear), so that it is difficult to find general conclusions. Two possible continuing 
sources of energy of the turbulence suggest themselves. The first is convection from 
sources of heat within the fluid, but there seems to be too little known about turbu- 
lence accompanying convection for consideration of this case to be useful. Another 
possible source is the mean velocity distribution. In the simple and interesting case 

* The argument here will be rougher than in the cases of rapid distortion. 



294 The effective pressure exerted by a gas in turbulent motion 

in which the mean velocity distribution takes the form of a differential rotation about 
an axis of symmetry, and the distortion has symmetry about the same axis, the mean 
velocity of each circular filament of fluid will change in the manner required for 
preservation of angular momentum (assuming that no appreciable change in the 
spatial distribution of mean angular momentum is brought about by means of 
turbulent exchange). We can also make use of the well-known empirical fact that 
the equilibrium magnitude of turbulent velocity fluctuations is proportional to the 
maximum mean velocity difference in the fluid, for boundaries of similar shape, and 
with the neglect of any effects due to change of the Mach number of the motion. 
Hence if the axis of symmetry is a principal axis of the strain, with extension ratio 
e 3 , we shall have for the change in the mean velocity 

U^iUXK ( ei = e 2 ), ....(7) 

and then since all three components of the turbulent velocity come to equilibrium 
with the new mean velocity distribution, 

V/K 2 )o = V/K 2 )o = % 2 /K 2 ) = e r 2 > 

and pUiUj ?& p u^Tj 

e i e 2 e 3 e l 

2/3 /p\ 5 /3 

( W*j)o- ( 8 ) 



f3f 3 It 



Po 

The behaviour of all components of the stress tensor is the same in this case, and the 
turbulent pressure varies with p and with the ratio e B /e v If the fluid is being contracted 
in the axial direction, with no change in the lateral plane (i.e. e x = e 2 = 1), the 
turbulent pressure is proportional to (p/p ) n an( i n = 1 ( m th* 8 case no change occurs 
in the mean velocity distribution), whereas if the fluid is expanding in the lateral 
plane without change in the axial direction, the index is 2. If the distortion is 
isotropic we have the value 5/3 for the index ; the same value was found for an iso- 
tropic distortion which takes place rapidly, so that it seems very likely that this will 
be the approximate value of the index for isotropic distortions of all speeds in the 
case of a rotationally symmetric mean velocity distribution. 

4. Application op the Results 

With the aid of results of the above kind it may be possible to analyse the history 
of masses of gas which are being subjected to distortion, for example, by explosive 
expansion or by gravitational contraction. Many factors other than turbulent pres- 
sure are involved, and a complete discussion requires a wider knowledge of astro- 
physics than the author possesses. One or two features of the problem are perhaps 
worthy of comment and may lead to further developments. It has been shown that 
in some cases the turbulent pressure varies with mean density, under adiabatic 
distortion, in the same way as for a polytropic gas, i.e. turbulent pressure is propor- 
tional to (p/p ) n . Such cases were special only in the sense that the kind of distortion 
considered was of a fairly simple kind, so that it seems reasonable to regard the 
qualitative features of such cases as having general significance. It was also found 



G. K. Batchelor 295 

that the value of the index n could be more than 5/3. 5/3 is the maximum possible 
value of the index arising from the thermal motion of the particles of the gas, and 
the effective value of the index for radiation is 4/3 (Eddington, 1927). Thus there 
exist cases in which the rate of increase of turbulent pressure with increase of mean 
density, as brought about by adiabatic compression of a specified kind, is greater 
than that of gas thermal pressure or of radiation pressure. In such cases, and pro- 
vided the turbulent pressure is not negligible (i.e. provided the Mach number of the 
turbulence is not small compared with unity), questions of equilibrium between 
gravitational force and pressure gradient will need to be decided with due account 
of the existence of the turbulent pressure. 

The case of a disk-shaped mass of gas which has a rotationally symmetrical distri- 
bution of mean velocity, and which is being flattened still further by gravitational 
self-attraction (the gravitational force in the plane of the disk being resisted by the 
centrifugal force of the rotation) is one of particular interest in that it may represent 
one part of the process of formation of galaxies, and possibly of other aggregates of 
interstellar matter. According to the analysis of this note, the value of the index 
(for the turbulent pressure) when further flattening of the disk occurs is 3 if the whole 
flattening process takes place more-or-less "instantaneously" (using the mean 
velocity of the fluid as a standard), and is 1 if the flattening occurs so slowly that the 
turbulence is in balance with the mean velocity distribution at each stage of the 
flattening. The order of magnitude of the duration of an axial contraction which is 
neither slow nor fast in the above senses will be given by the time required for an 
element of fluid in a typical position to make one circuit round the axis. Possibly 
some of the axial contractions that occur as a result of self-gravitation are closer to 
being "rapid" than to being "slow". If so, the index n will approximate to 3, and 
will surely be greater than 5/3. If the values of the index n for the turbulent pressure 
and the index y for the gas thermal pressure are widely different, there will be a 
strong tendency for the equilibrium state of the disk of gas to be such that the two 
pressures are of the same order of magnitude, for this is the state at which further 
contraction would generate a very large turbulent pressure which in turn would resist 
the contraction. Thus there may exist a tendency for flattened disks of gas to settle 
down to a state in which the root-mean-square of the turbulent velocity fluctuations 
is of the same order as the thermal velocities in the gas. The occurrence of a Mach 
number of the turbulence in the neighbourhood of unity may have some general 
significance, if these ideas are correct. 



References 

Batchelor, G. K. and Proud man, Ian . . . 1953 "The Effect of Rapid Distortion of a 

Fluid in Turbulent Motion", Quart. J. 
Mech. appl. Math., 6. 

Eddington, A. S 1927 The Internal Constitution of the Stars. 

University Press, Cambridge, p. 29. 

Ribner, H. S. and Tucker, M 1952 "Spectrum of Turbulence in a Contracting 

Stream". National Advisory Com- 
mittee for Aeronautics, U.S.A., Tech- 
nical Note No. 2606. 



"Turbulence", Kinetic Temperature, and Electron Temperature 
in Stellar Atmospheres* 

P. L. Bhatnagar, M. Krook, D. H. Menzel, and R. N. Thomas 

Harvard College Observatory, Cambridge, Massachusetts, U.S.A. 

Summary 

The phenomenological use of the term "astronomical turbulence" is reviewed and earlier conclusions 
that the physical nature of the phenomenon is more likely anisotropic mass-motion, or jet-prominences, 
than the customary aerodynamic turbulence are restated. The primary problem under such conditions 
is the relative importance of mechanical energy-transport and momentum transport in perturbing the 
structure of the atmosphere. The problem of the difference between kinetic temperatures of the atoms 
and electrons is treated, and it is concluded that the difference is negligible in those parts of the stellar 
atmosphere which are in a statistically-steady state. 



1. Introduction 

Theoretical investigations of the outer layers of a star have been based, for the most 
part, on the "standard" model of a stellar atmosphere. The following assumptions 
specify this model : 

(Al) Radiative equilibrium. 

(A2) Hydrostatic equilibrium of the matter. 

(A3) Local thermodynamic equilibrium for the matter. 

(A4) Neglect of electric and magnetic fields. 

The physical structure of a standard model atmosphere is then determined com- 
pletely (at least in principle) when its effective temperature, surface gravity, and 
chemical composition are specified. In particular, these data determine unambigu- 
ously the density and temperature distribution for the model. A prediction of the 
spectrum follows from additional assumptions on the mode of fine formation. The 
term "turbulence" has been used in a rather vague way to characterize certain dis^ 
crepancies between actual observations and predictions from the model. 

At least six types of observational results provide the background for the discussion 
of "turbulence" in stellar atmospheres. These are: 

(Tl) The extent of the atmosphere. 

(T2) The density distribution. 

(T3) A random, line-of-sight velocity deduced from the so-called Doppler 
segment of the curve of growth. 

(T4) A random, line-of-sight velocity deduced from measures of line profiles. 

(T5) The directly observed, or the inferred, presence of some form of promi- 
nence activity. 

(T6) The occurrence of anomalously high excitation; this feature deserves 
more attention than it has received to date. 



* The research reported in this document has been made possible through support and sponsorship extended by the Geo- 
physics Research Division of the Air Force Cambridge Research Centre, under Contract No. AF19(604)-146. It is published for 
technical information only, and does not necessarily represent recommendations or conclusions of the sponsoring agency. 

296 



P. L. Bhatnagab, M. Rrook, D. H. Menzel, and R. N. Thomas 297 

As regards (Tl-4), the significant features in the present context are: 

(i) The existence of atmospheres considerably more distended than in the 
standard model (Tl, 2). 

(ii) The existence of random velocities other than, and in addition to, thermal 
ones predicted by the standard model. 

What astronomers have usually called "turbulence" in a stellar atmosphere 
consists of a purely phenomenological interpretation of the residual random-velocity 
fields obtained in (T3, 4) when the standard-model thermal motions are subtracted 
out. No detailed model of an atmosphere actually containing such velocity fields 
has so far been presented. In a rather more qualitative way, "turbulence" is also 
invoked to explain the anomalous distension of an atmosphere (Tl, 2). 

The physical self-consistency of such atmospheric velocity fields has not as yet 
been investigated. Thus far investigations have not gone beyond an attempt to 
apply the statistical theory of isotropic turbulence (in the aerodynamic sense), 
to macroscopic motions in a stellar atmosphere. However, the conviction appears to 
be growing that a system of anisotropic mass-motions, or jet-prominences (T5), 
represents astronomical "turbulence" more closely than does the ordinary concept 
of aerodynamic turbulence. 

In discussions of astronomical "turbulence" it is common practice to neglect the 
effects of the coupling between the macroscopic turbulent velocity field and the 
microscopic thermal motion. The energy of the turbulence dissipated in this way 
has implicitly been assumed to be negligible. The few exceptions to this remark 
have considered the possibility that the coupling between turbulent motion and 
thermal motion may be significant only in the outermost layers — chromosphere or 
corona — and hence could not easily be observed. Most generally, it has been assumed 
that the existence of a distended atmosphere does not imply the existence of appre- 
ciable temperature anomalies in the regions responsible for the formation of 
absorption lines. 

Most studies of "turbulence" in stellar atmospheres depend on an assumption of 
isotropy. We believe that this assumption is unjustified and oversimplified. One 
should consider carefully and in detail the source of the astronomical "turbulence" 
in order to relate it to the conventional turbulence observed in physical laboratories. 
Physically, the onset of turbulence is associated either with thermal instability in a 
quiescent medium, or with dynamical instability in fluid flow. The earliest suggestion 
(Rosseland, 1929) of the stellar occurrence of hydrodynamic turbulence was in 
connection with stellar rotation, the high Reynolds number of the flow in the 
atmosphere leading to dynamical instability. On the other hand, turbulence associ- 
ated with the star's convective zone results from thermal instability. From the 
standpoint of mechanisms for generating turbulence, we note then that the axes of 
preferred motion are at right angles in these two cases. Consequently, we believe 
that the general problem should eventually be investigated from the standpoint of 
origin. 

In preceding discussions (Thomas, 1947, 1948, 1949) one of us has suggested that 
astronomical "turbulence" consists of strongly anisotropic mass-motions, preferen- 
tially in the radial direction, associated with high kinetic temperatures induced by 
this mass-motion. Any such system of mass-motion will produce atmospheric 
distention. If energy transfer predominates over momentum transfer, the distension 
will effectively result from a high kinetic temperature. On the other hand, for 



298 



"Turbulence," kinetic temperature, and electron temperature in stellar atmospheres 



sufficiently large momentum transfer, the distension results primarily from dynamic 
support. We believe that the Wolf-Rayet atmosphere represents an extreme case 
of the dynamic support. The solar chromosphere-corona forms an example of a case 
where the relative importance of momentum and energy transfer is not yet clear. 
Over a period of years, another of us has urged that a prominence model of a stellar 
atmosphere deserves more attention (Menzel, 1930, 1931, 1939, 1946). If small 
jet-prominences comprise essentially the entire atmosphere, momentum transfer 
will be more significant. However, if the jets eject into a gaseous medium, to which 
they supply energy, then energy transfer may be the more important. One may well 
question whether such a system of prominences could exist without creating a gaseous 
substratum. Then, given such a substratum, even if the main process is momentum 
transfer, one must investigate the extent to which the coupling of a jet-prominence 
with the medium may produce a high kinetic temperature. 

The translational kinetic energy of a jet-prominence is converted into thermal 
energy of the atmosphere by elastic collisions between jet particles and atmospheric 
particles. A jet atom loses its excess kinetic energy after a few collisions with 
atmospheric atoms. On the other hand, because of the small ratio of electron mass 
to atom mass, a jet atom would require many more collisions with atmospheric 
electrons to lose its excess kinetic energy. Thus the jet loses its translational energy 
by raising the kinetic temperature T k of the atmospheric atoms (which, hereafter, 
we call Process a). The elastic collisions of atmospheric electrons with atmospheric 
atoms and ions then tend to raise the electron temperature T e (Process b). This 
last process competes with such processes as radiative capture, collisional excitation 
with ultimate radiation, which tend to lower the electron temperature (Process c). 
The question then arises whether a steady-state atmosphere with such mass-motions 
could exist in which the atom kinetic temperature T k is significantly greater than the 
electron kinetic temperature T e . 

From the observational aspects, the above picture would be clarified if the atmo- 
sphere kinetic temperature, T k , could be measured. Working within the "standard" 
model, one customarily assumes that T k is equal to the electron temperature, T e , 
and identifies the latter with the excitation temperature, T ex . The uncertainty in 
this last identification has been discussed (Thomas, 1949). In the solar chromosphere, 
however, the largest current source of uncertainty seems to be the discrepancy 
between T k and T e . T k is largely fixed by optical observations. T e comes from radio 
observations. It appears that T e < T k . We must, however, always remember that 
the radio and the visual observations refer to very different layers of the solar 
atmosphere, and the comparison of T k and T e is by extrapolation. 

Criticism has been directed at the eclipse observations of line profiles (Miyamoto, 
1953), largely because self-absorption may cause the observed profiles to appear 
wider than the true Doppler profile at temperature T k . We believe it significant that 
these same line profiles place an upper limit on the tangential component of any 
turbulent velocities. This limit lies considerably below the value required for the 
interpretation of the chromospheric density gradient as arising from an isotropic 
turbulence. Thus we return to the question whether a purely radial set of motions 
can exist without raising T k . We have attempted, as yet unsuccessfully, to investi- 
gate how a departure from Maxwellian velocity distribution for the electrons may 
affect the value of T e inferred from the radio observations. The question then arises 
whether there could actually exist a difference between T e and T k . 



P. L. Bhatnagar, M. Krook, D. H. Menzel, and R. N. Thomas 299 

The literature on discharge-tube phenomena contains reference to situations with 
T k < T e . In investigating the structure of very strong shock-waves (non-steady 
state), we have encountered a situation with T k > T e . The question then arises 
whether T k > T e might occur in a steady-state in a stellar atmosphere. 

2. Investigation of the Possibility of the Steady-state 
Configuration T k > T e 

With reference to the processes of energy transfer mentioned above, a balance of 
process (a), energy transfer from atoms in a jet-prominence to those in the atmo- 
sphere, against process (b), energy transfer from atmospheric atoms to atmospheric 
electrons, should give the atom kinetic temperature in terms of the energy dissipation 
from the mechanical energy source. A balance of (b) against (c), energy dissipation 
by the electrons, should give the above discussed comparison between electron and 
atom kinetic temperatures. The "kind" of energy dissipation, (a), to be expected 
from a system of jets, or prominences, has previously been discussed in a rough 
manner for the solar spicules (Thomas, 1948). Here, we consider the T e , T k distinc- 
tion. We consider a hydrogen atmosphere in the following. 

Process 6 : elastic collisions in the medium transfers energy from the atoms to the 
electrons at the rate : 

TYt 

E b = a h — e N e N a v e k(T k - T.), .... (1) 

m a 

where the subscripts a and e refer to atom and electron, respectively; where N is 
the particle density; m, the mass; v e the mean electron velocity; T e the kinetic 
temperature of the electrons and T k of the atoms ; and k, Boltzmann's constant. 

We have derived the cross sections a b , for energy transfer in elastic collisions, for 
the respective cases of neutral atoms and of ions interacting with electrons. We 
shall omit the details of the derivation in this paper. 

Atom-electron : 

a b = 8 /! \i + Vtl TA V2 a „ 4.10-™, • • • • (2) 

b V 3tt L m a T J 

where a is the ordinary kinetic cross-section, of the order of 10~ 16 cm 2 . 
Ion-electron : 

y2rr / rn T \ ~ 3 / 2 

T I 1 + m TJ ( ^ )-2£4 ln [1 + i^T e e^N e -^] 

c^ 1-2 . 10- 4 T 6 - 2 [1 + 0-13 ln T e N- l, % .... (3) 

where s is the electronic charge. 

Process c: the electrons lose energy in three ways. (1-c): by interaction with the 
ions they may show an excess of emission over the absorption of radiation in free-free 
(ff) transitions. Defining: 

X = ^; P 1 = 3-2.10^kN i N e T e -^. ....(4) 



300 "Turbulence", kinetic temperature, and electron temperature in stellar atmospheres 

We obtain the rates of loss and gain by these two processes (Menzel, 1937) : 

IcT 
emission = E ff = P-, —r-^ .... (5) 



absorption = A ff — E ff .W.X 



where B is the Rydberg constant, h is Planck's constant, and the subscript i refers 
to ions. In the derivation of (5) we have taken the incident radiation field to consist 
of black-body emission at a temperature T r , with a mean dilution factor W. 

(2-c) : the electron component of the medium loses energy by free-bound and gains 
energy by bound-free transitions (Menzel, 1937): 

emission = E fb = P x n~ z — F fb E n , (7) 

00 

absorption = A bf = WP x n^X n e^Y n ^ 2 e- r l& w a-i - (a + A)-*] - F bf E n 

a = l 

....(8) 
where the F's represent numbers of transitions per cubic centimetre per second : 

F fb E n = P in -*X n e*-E t ( - X n ), .... (9) 

F bf E n = P x n-*X n e x » | {b n E t (- aY n ) - E,{- Y n [a + A])} (10) 



a=l 



These formulas refer to hydrogenic atoms, with n as the principal quantum 
number : 

E n E n E x 

Xn = W' Yn = k¥~' En = '' n*' El== hE ' 

The quantity b n specifies the degree of departure from thermodynamic equilibrium. 

where a> n = 2n 2 

the statistical weight of level n. We have introduced the exponential integral, 

too 
x~ x er x dx. 

(3-c) : inelastic colhsions represent the third process of energy loss by the electrons. 
These colhsions may ionize the atom or merely excite it. The net energy loss in 
ionization processes is (Matsttshima, 1952): 

C ni = G ni E n = 3-6 . lO-UPjJW^-U - K^)X n (l + *»K0»« • • • -(H) 

where the subscripts ni refer to the ionization from level n. Q ni is the mean cross- 
section for ionization, expressed in atomic units. For collisional excitation from 
level n to n" and the reverse, we have the net loss (Matstjshima, 1952): 

^nn" ^nn'K^n ^ n") ^ nn"*^ nn" 



3-6 . IQrup^TW b n ( 1 - b f} e*-(l + X nn .)X nn „Q nn .u n 



(12) 



P. L. Bhatnagar, M. Krook, D. H. Menzbl, and R. N. Thomas 301 

Thus, equating energy gains by process b to energy loss by processes c, we have 

E b = (E„ - A„) + l[(E fb - A hf ) + C ni + I C nn „] (13) 

n n">n 

We now introduce the conditions that the atmosphere be in a statistically steady 

state, an assumption requiring that N n , N it and N e be constant. Given T e , T n , W, 
and N e , we may employ the constancy of N n to determine the b n . Constancy of the 
numbers of ions and electrons supplies the auxiliary condition : 



2(C ni + F bf - F fb ) = 

n 

balance of the total number of transitions up and down. 
If, now, we set: 



17 * 



-fb 



Ffb E n ; A b 



l bf 



Fbf E n 



and use (14), (13) becomes: 

E b = (E„ - A„) + 2L(Ef>* - A w *) + E Xn {F fb - F bf - C ni ) + 2 C nB .] 

n n">n 

Introducing the explicit forms, we obtain : 

a b -^N e N a v e k(T k ~T e ) 

='.s[-^+(*-ry+(*-dn)---): 

+ P>!\n-%\ - WX n Y n ~H x » I e^HK^ 1 - [a + X]-*)] 

n a = l 

+ X ln e^E i (-~X n ) 

[i - ErH- x n ) I {M?,(- «rj - ^(- r> + a])}] 

a = l 

+ X 11t . 3-6 . 10rUN e T B 1 l*a>M- (1 - 6„- 1 )(l + X n )Q ni 

+ 2 (i - K • ^(i + z^)X M .x ln -^"'G nil .]]. 

Now, employ (4), introduce 
and insert numerical values : 

T e =T k - 8-7 . lo-^-^-w^a-Kfra . io-«r e [i - . . .] 

+ 2[^- 3 (i -...) + 7-2 . io-«#x 1/2 *i«»*«(< io)]} 



.(14) 



.(15) 



(16) 



(17) 



Consider now the term <7 6 -1 iV r i ^ r a _1 . This term represents elastic collisions of electrons 
with both ions and neutral atoms. Thus, by (3) and (4) : 



o^N<N-i 



I er 6 (ion) + cr & (atom) -^ I 

= [1-2 . 10~ 4 (1 + 0-13 In T e N~ xl *) + 4 . IQ-^N/T-^^K^'Y^T^ 

....(18) 



302 "Turbulence", kinetic temperature, and electron temperature in stellar atmospheres 

where we have used the dissociation equation in the form modified for departure 
from thermodynamic equilibrium : 

[to ~| 3/2 

wd r b ^- •••• (i9) 

For the solar chromosphere, we can set the limits : 

5-10 3 < T e < 10 5 ; 10 10 < N e < 10 12 

and all computations (Matsushima, 1952) give b x < 10 6 ; 6 30 -->-< 1. Observations 
show the effective hydrogen continuum to begin at about H 35 . The excluded-volume 
principle allows the summation in the partition function to be terminated at this 
point. In such cases the ion-electron term in (18) always predominates. The range 
of values specified above fits the inequality : 

Further, the bracketed quantity in (17), in general, does not exceed 10. Thus: 

T k -T e < 10-»ZV 

This analysis thus leads to the significant fact that the difference between T e and T k 
is indeed negligible. This result appears to remove all possibility of reconciling the 
radio and optical observations in terms of differences between kinetic and electron 
temperature, at least for the case of a homogeneous atmosphere in a statistically- 
steady state. One further possibility of reconciliation lies in the investigation 
mentioned earlier of the influence upon the free-free emission of a non-Maxwellian 
velocity distribution for the electrons. Another, lies in an atmosphere with marked 
non-uniform distribution of temperatures, such as the models proposed by Giovanelli 
(1949) and by Hagen (1953) of a hot chromosphere containing cool columns of gas. 

3. Comment on the Result T e ~ T k 

In the literature on electric discharge in gases one finds frequent reference to situa- 
tions in which T e > T k . Physically, our above conclusion, that T k ~ T 6 in a steady 
state, is a consequence of the fact that in electron-atom collisions the elastic cross- 
section is much larger than any of the inelastic cross-sections. Consequently, to 
obtain a steady state with T e > T k , an efficient mechanism for the atom to dissipate 
energy must be introduced. In the discharge tube, wall-effects may provide such an 
energy sink. However, this mechanism must be explicitly introduced, and the rate 
at which it dissipates energy must be calculated. Thus, for example, Alfven (1950) 
balances the energy gained by an electron drifting in an electric field against its loss 
in elastic collision with ions and atoms. The possibility of a steady-state is simply 
assumed, and the conditions then holding are determined by assuming the above 
energy balance to be exact. No mention, however, is made of the mechanism of 
energy loss by the atoms. Unless this mechanism is introduced, the assumed situation 
clearly violates the conservation of energy. 



References 

Aleven, H. . 1950 Cosmical Electrodynamics, p. 45 (Oxford 

University Press). 

Giovanelli, R. G 1949 M.N., 109, 298. 

Hagen, J. P 1953 Private communication. 



P. Wellmann 303 

Matsushima, S 1952 Ap. J., 115, 544. 

Menzel, D. H 1930 Amer. Astron. Soc. Abstracts, 6, 370- 

1931 Popul. Astron., 39, 16. 

1937 Ap. J., 85, 330. 

1939 Popul. Astron., 47, 6, 66, 124. 

1946 Physica, 12, 768. 

Miyamoto, S 1953 Private communication. 

Kosseland, S 1929 M.N., 89, 49. 

Thomas, R. N 1947 Ap. J., 52, 158. 

1948 Ap. J., 108, 130. 

1949 Ap. J., 109, 500. 



The Intensity of Emission Lines in Stellar Spectra 

P. Wellmann 

Hamburger Sternwarte, Hamburg-Bergedorf, Germany 

Summary 

The conditions in an extended stellar atmosphere may be adequately explained by the theory of diluted 

radiation if it is postulated that the optical depth in the lines is small enough. 

In order to check this basic postulate the dependence of emission intensities on the numbers of atoms 
is developed by analogy with the curve of growth for absorption lines. This relation, called curve of 
growth for emission lines, is used in analysing observed emission spectra and reveals that the assumption 
of small optical depths is permissible in many cases. 

Moreover, it is seen that even a strong emission spectrum may be affected by the shell absorption. 
If these two phenomena, i.e. spectra, are treated as a unit, the curves of growth for absorption and 
emission lines appear as limiting cases of a more general relation between line intensity, transition prob- 
ability, and characteristic data of the atmosphere. This relation describes the difference between various 
shell spectra and the changes in the spectrum of an expanding atmosphere, and it can be used to study 
the structure of a shell. 

A generalized equation of ionization has to be introduced for the investigation of emission spectra. 



1. The Origin of Emission Lines 

It is now almost unanimously recognized that emission lines visible in the light of a 
star as a whole are caused by the action of diluted radiation in extended atmospheres. 
The observations of the solar chromosphere, of Be-stars, WR-stars, and novae, 
prove that the monochromatic emission is generated in the outer low-density parts 
of the stellar envelopes, and it is obvious that on the average the more extended 
atmospheres have the stronger emission. 

Formerly the presence of emissions comparable in intensity with the continuum 
was attributed entirely to the relatively large mass of gas in the shells. This explana- 
tion implicitly presupposed normal excitation and local thermodynamical equili- 
brium. In this case, the flux of continuous radiation in a line frequency may suffer 
absorption and scattering after emerging from the deeper parts of the star. At any 
rate it is diminished by absorption. Under the most favourable circumstances, the 
radius of the star being infinitely small, the scattering does not influence the total 
energy output. But normally a fraction of the back flux of scattered light will be 
re-absorbed in the interior of the star and this part is also lost in the line frequency. If 
we assume radial symmetry of star and envelope, absorption lines always result irre- 
spective of the extent and mass of the atmosphere. Therefore, if an emission fine exists, 



304 The intensity of emission lines in stellar spectra 

some energy has been transferred from one frequency to another by monochromatic 
processes. This rather general argument proves that the principle of detailed 
balancing is no longer valid, and that it must be replaced by the more general 
principle that a steady state exists. We write down the steady-state conditions 
for a gas consisting of atoms with n energy levels and m allowed transitions between 
these levels, the atoms being in a radiation field of density u(v). Let the population 
of a given level be N atoms per cm 3 , then we have 

dN , du(v) 

— = and — - — =0. (1) 

dt dt v ' 

The members of the first set of equations have the form 

2NiB ijUij + 2NMm + B kj u kj ) = N^iAji + B H u it ) + I,B jk u ik \ ... .(2) 

i k i k 

where A , B = transition probabilities ; i, k = levels lower or higher than level j. 
Only n — 1 of these equations are independent of one another. They contain 
n — 1 ratios NJN^. because they are homogeneous in N. But they contain also as 
unknowns the functions u(v). The equations of the second set are equivalent to the 
equations of transport of radiation. Explicitly written they run 



cos 6 — ^ = — 

ds 4tt 



NiB ik — I ik + N k (A ki + B ki u ki ) 
c 



...(3) 



hi 



with u ilr — - I I ik dco. 

The absolute values of the N's now appear as unknowns, and we have to add the 
condition 

2^- = N t 

i 

that the total number of atoms is final. 

There are n + m, equations and n + m unknown functions. On principle it is 
possible to solve the problem. These considerations are valid in the same form if 
continuous states are included. The continuum will be divided into elements of 
infinitely small width dE which will be treated as an infinite number of discrete 
states. This consideration also holds for a mixture of different atoms and molecules. 
Furthermore, the result is the same if collisional excitation is introduced in (2). 
The solution is completely independent of any concept of temperature. The distribu- 
tion with frequency of the radiant energy and the energy distribution among the 
free electrons are not assumed, but are deduced from the theory. 

The practical solution of this system of integro-differential equations is extremely 
complicated, but we can reduce the problem to simpler forms by adequate 
approximations. 

It has been proved that the population of the continuous states, that is the 
frequency of electrons with certain velocities, is very well described by the Maxwell 
distribution, even when the departures from thermodynamic equilibrium in the 
populations of the discrete states are rather strong. By introducing the Maxwell 
law an electron temperature is defined and determined. 



P. Wellmann 305 

The equation of radiation transport can be transformed into 

u(v) = ~ I*e- T dco + - ee- T dsdco, r = K Nds, (4) 

C j(t> C J(o Js Js 

where e is the energy emitted by the unit of volume in the shell. /*, the intensity 
of radiation at the photosphere, is given as a boundary condition. It is customary 
to write 

2hv z 1 

I*(v, ») = -r • -± = I P {v), 

ekT, — 1 

i.e. to neglect the darkening to the limb of the star and to introduce a radiation 
temperature T s . The next simplification implies a significant loss of generality. 
Instead of using the transport equations (3), we substitute in (4) an appropriate 
assumed functional form. 

First it is assumed that the second term is negligible in consequence of the low 
density in the shell. Hence 

u(v) = ^[ e -~— = u P (v) . W = u P (v) . «T . — , .... (5) 

C Ju) 4-7T 477 

a>* is the solid angle subtended by the star ; 

W a = —™( — ) ....(6) 

9 4t7 \2rJ V ' 

is the geometrical dilution of the radiation from the star, e -? is the dilution by 
absorption in the medium between the star and the point in question. In most 
computations of the populations of discrete levels, however, W is assumed to be 
constant, i.e. only the geometrical part is considered. This is permissible if the 
optical depth of the atmosphere is small in all wavelengths. In some cases W was 
assumed to be a certain constant at wavelength longer than the Lyman limit, and a 
constant considerably smaller below this limit, in order to allow for a strong absorp- 
tion in the Lyman continuum. 

If we introduce arbitrary functions we are no longer sure that the solution is 
correct with regard to the energy balance. Therefore it is necessary to introduce the 
condition of constant energy content. This yields a relation between T e and T s 
(Baker, Menzel, Aller, 1938; Woolley, 1947)*. 

2. The Population op Different Atomic Levels 

The problem is now reduced to a rather simple form: a gas of temperature T e is 
excited by a radiation of temperature T s , the radiation being diluted by the factor W. 
How are the populations changed relative to the equilibrium at T e ? 

The first treatment of this problem was published by Rosseland (1926) in his 
classical papers on the theory of cycles. He began with three discrete states and 
showed that the transition 1 — 3 from the ground state to the highest level was not 
compensated by 3 — 1, but partly by 3 — 2 — 1, and vice versa. The probability of the 
cycle 1 — 2 — 3 — 1 is smaller than that of the reverse cycle by a factor equal to W. 



* Strictly, the results of the next section should be known in establishing the relation between T e and T 3 . But it is actually 
possible to solve the problem in advance with sufficient accuracy by a process of successive approximations. 



306 The intensity of emission lines in stellar spectra 

This means that the emission in the two lines of longer wavelength was strengthened 
considerably at the expense of the line of shortest wavelength. In most cases W is 
very small: therefore one of the cycles nearly ceases to operate and the lines of 
longer wavelength appear as pure emissions. Since this work by Rosseland it has 
been possible to understand the existence of emissions. Further application of his 
method showed that the first results give a qualitative guide for many other cases. 
The steps of small energy difference always gain emission at the expense of transitions 
of larger or of the largest energy difference. The emission caused by recombination is 
a limiting case. It is qualitatively understood on the same basis why in a series the 
emission becomes weaker with increasing quantum number. 

Another important rule follows from these early computations of dilution effects 
by Rosseland (1926), Ambartsumian (1933), Strtjve (1935a), and other authors. 
When the ground state combines with higher levels, the populations of these levels 
are reduced, approximately by a factor W, relative to that of the ground state. But, 
if one of the higher states is metastable, its population is nearly independent of the 
dilution and bears the normal relation to that of the ground state. This holds 
approximately in more complicated cases with a larger number of levels if one or 
more metastable levels are included. These metastable levels reduce the effects of the 
cycles ; their high populations cause a relative strengthening of all the fines due to 
transitions starting from these levels. This is the key to the explanation of a great 
many observations. 

In special cases this general, qualitative reasoning has been replaced by numerical 
solutions of the steady-state conditions assuming a larger number of energy levels 
with individual values of frequencies and transition probabilities. 

This has been done by Cillie (1936) and by Menzel and his collaborators for 
hydrogen, all atomic constants of which are known functions of the quantum numbers 
(Menzel, 1937; Menzel, Baker, 1937; Baker, Menzel, 1938; Menzel, Aller, 
Baker, 1938). The solution of the complete infinite system of linear equations (2) 
is given in series development. The results are relative populations NJN or certain 
coefficients b { , defined by Menzel in the following way. 

In thermodynamic equilibrium the number of atoms in the state * may be written 

CO Jb ■ ^* 

AT — i- N + N p* T > 

i0 ~ 2w+ ° e (2rrmkT e f* ' 

since the Saha equation is valid. In the general problem the departure from this 
expression is measured by the factor b, where 

tf< = Mr<o. N+ = b k N +. 

Since the system is homogeneous in N and in b, the 6-values may and must be 
normalized. This is done by putting b k = 1 for the ion such that 

The natural definition would perhaps have been to introduce 6=1 for the ground 
state, but Menzel's definition has many advantages for practical computation. 

Menzel and his collaborators calculated the 6's for the first fifteen levels of 
hydrogens as functions of T s and T e for the limiting case of very low W or very 



. P. Wellmann 307 

strong dilution. In this case bj^ is approximately proportional to W, hence all the 
6/s except b ± are independent of W, and 6 X is proportional to \jW. 

The values of 6 for He u, valid for T s and T e , are the same as the hydrogen values 
for TJ4: and TJ4. 

The 6-values of He i have been calculated, but it is not possible here to use serial 
laws in order to establish a manageable infinite system. Hence Strtjve (1935b), 
later Strtjve and Wtjrm (1938), used only six levels. Goldberg (1941) used eight 
levels and improved the method by introducing integrals as an approximation to the 
sum of the higher states, but he restricted the solution to W & 0. The present author 
used Goldberg's method and computed the 6-values of thirteen levels for different 
W and of twenty-one levels for W (*a (Wellmann, 1952a). The introduction of the 
higher levels up to the quantum number 4 enables us to deal with some of the 
important He i-emission lines in the visual and photographic region of the spectrum. 
Furthermore, the extension of the work to larger W made the results applicable to 
atmospheres of intermediate heights and to the lower parts of high atmospheres. 
The He-spectrum is particularly interesting. Because of the two term systems and 
the two metastable levels, it shows many lines having very different behaviour. 
This makes He more important than hydrogen, the lines of which are too uniform in 
their reaction to dilution and temperature. 

The results may be described by some general rules : 

(1) The relative population of two levels is nearly independent of dilution: the 
population of the ground state and the metastable 2 1 #-state. 

(2) The higher singlet levels are depopulated proportionally to W. 

(3) The population of the metastable level 2 3 $ gains a factor of from 10 to 20 relative 
to the ground state, but is independent of W if the dilution exceeds a certain limit. 

(4) The population of the higher triplet states is proportional to W. Most of the 
triplets behave like the singlet states, except the w 3 P-levels in which the atoms are 
ten to twenty times more frequent. 

3. The Curve of Growth for Emission Lines 
The 6-values furnish the basis for the construction of the curve of growth. When we 
derive a curve of growth for absorption lines, we plot a certain function of the line 
intensity against a theoretical intensity, which depends on the oscillator strength of 
the line and the population of the lower level given by the Boltzmann distribution. 
We expect the emission intensity, the oscillator strength /, and the 6 of the upper 
state, to be related. But, as emission does not exist without more or less reabsorption, 
the 6-value of the lower level will also affect these functions. 

In the following we have to remember that we are only trying to obtain a first idea 
of the relation. We have a complicated theory of the formation of absorption lines, 
but in dealing with absorptions in integrated starlight, use was generally made of an 
interpolation formula for the profiles (Unsold, 1938; Menzel, 1939) and only in 
more recent papers the theory of atmospheric models has been considered ( Wrubel, 
1949, 1950, 1954). To the same degree of first approximation, we regard the emission 
of a homogeneous gas filling a cylindrical volume of height H. The intensity directed 
outward at the front surface is 

r f [ H n 2hv * L , w Xcbi r r N k l 7 



308 The intensity of emission lines in stellar spectra 

Assuming N i>h to be constant, we get 

C* 'co.N,' v c* b k ' > 

f 00 

with A v = \ (1 - e- KNtB )dv, 

and after re-arranging 



_*^= const.-™. ....(8) 

v 4 Co,- N h v 



'i x * k 

The theory of absorption gives a relation between 

Av NJH 

— and , 

v v 

the well-known curve of growth for absorption in an absorption tube. 
Because of (8) this may be regarded as a relation between 

L^.li^ d Mi, .... (9 ) 

and this is the required curve of growth for the emission intensity. By this procedure 
we avoid the repetition of laborious integrations already discussed in the theory of 
absorption lines. The resulting curve is easily interpreted. The linear part expresses 
the fact that the emission is proportional to the transition probability and the 
number of atoms in the upper state, as long as the optical depth is small. When 
it becomes larger the reabsorption increases sensibly and causes a bending of the 
curve. The reabsorption limits the intensity in the middle of the emission profile to 
W . I p . When the number of atoms grows further, then only the width of the line 
can increase and the intensity follows a square-root law. 

As an example of a Be star with strong emissions of H and He i, y Cass has been 
investigated (Wellmann, 1952c). From the curve of growth, constructed according 
to (9), the values T s ~T e ~ 10,000°K and W ~ 10~ 2 were derived. The figures 
are not defined very precisely, but as a first attempt to deduce these values from 
emission lines they may be sufficient. The Balmer lines fall on the linear part of the 
curve, as do most of the He lines, but some He lines in the upper part show the 
beginning of the bending. Hence we conclude that the lines do not suffer from 
appreciable reabsorption: some He lines with very strong population of the lower 
states are the only ones to be slightly affected. The optical depth is, therefore, small 
for most of the emissions, though the whole atmosphere participates in producing 
them. This results from the strong Doppler broadening of the fines. 

The emission He I A3888 is exceptional in that it is too weak compared with the 
prediction of the curve of growth. This is the strongest transition between the 
metastable level 2 3 # and the higher triplet states, and the extremely high population 
of this level which results from the dilution will cause a very strong absorption in the 
fine A3888. In the spectra of shells with moderate height showing weak emissions, 
this line is often the only line which remains when the normal absorption spectrum 
is blotted out by dilution. However, the reabsorption is already allowed for in the 
curve of growth and the observed reduction must be the effect of another superposed 



P. Wellmann 



309 



absorption spectrum. That is easily explained by the absorption of the light of the 
star in the atmosphere. 

4. The Complete Shell Spectrum of a Be-star 

We have now to change our theoretical assumptions by including the shell absorption 
spectrum. This is at first sight a serious matter because we are apparently forced 
to seek the solution of a problem of radiation transfer. But fortunately the re-absorp- 
tion is negligible, and we thereby escape all the difficulties. The equation of radiation 
transport reduces to a simple expression, and immediately leads to a formula for the 
flow of radiation in the emission lines at the outer boundary of an atmosphere with 
radial symmetry. We get, assuming a homogeneous envelope (Wellmann, 1952b), 

E = «.'hv{b k D kin - b t D ilc )N+N e V, 

with the function r}{ W) according to the following table : 



w . 


1-0 


0-5 


0-10 


001 





V ■ 


0-5 


0-5 


0-81 


0-97 


1 



The product b t . D ik is the frequency of the transition i -> k. E is the energy 
expressed in equivalent width, positive for emission, negative for absorption. V is 
the volume, a' a certain constant. 

After some transformation we have 



E 



0)v 






const. 



NJH 



y>, y = i — 






If no shell absorption is present, as in cases of extreme dilutions, b t . D ilc vanishes, 
ip = + 1 and the result is the linear part of our curve of growth for pure emission. 
The other limit is given by thermodynamical equilibrium, W = + 1, 7] = 0-5, 



b f = h = 1, D a 



D w , hence 



W 



1, and the result is the linear part of the 



normal curve of growth of absorption lines. Imagine a low atmosphere. It will 
show the normal absorption spectrum. If this atmosphere expands for some reason, 
the dilution factor diminishes and ip grows. Consequently, the absorption lines are 
diminished by superposed emission. If the expansion is large enough, the absorption 
will be masked completely, later the emission will dominate and, finally, when 
y> = + 1 is reached, the emission will show the undisturbed maximum value. 

The growth of ip with dilution, however, differs from line to line. The lines He i 
/16678, A5875, and A3888 are examples of a normal singlet line, a normal triplet line 
and a line of which the lower state is metastable. With T s = T e = 25,000°, we find 



X 


W 


1 


0-1 


0-01 





6678 
5875 
3888 


- 1 

- 1 

- 1 


+ 0-65 
+ 0-09 
- 015 


+ 0-96 
+ 0-59 
+ 0-09 


+ 1 
+ 1 
+ 0-13 



We see that in an atmosphere with moderate dilution ( W from 0-5 to 0- 1 ) the singlets 
have the strongest tendency to emission, the triplets have weaker emission or 



310 



The intensity of emission lines in stellar spectra 



stronger absorption, while A3888 is still an absorption line. Even with extreme 
dilution, A3888 remains fainter than the curve of growth for pure emission predicts. 

This is, at least on principle, what we observe in the spectra of different shell 
stars or in the spectra of stars with outer atmospheres of variable extent. 

Finally, this formula not only describes the mean quality of an atmosphere, but 
helps in attacking the problem of stratification. If we want to allow for the variability 
of density and dilution with distance from the star, we must write : 



E = 47ra'/w 



/* oo 



(MW - b t D a )N+Njr*dr. 



The function in brackets is given by the theory and differs from line to line. If 

N+N e =f(r, ai ,a 2 , . . .) 

is a given function containing n unknown parameters a, it is possible to deduce 
these parameters from the measured intensities of n lines. Because not all of the 
lines are suitable for actual use, the number of parameters is limited, but we can 
secure preliminary information on the structure of such an emitting atmosphere. 
The first tentative calculations show that in the case of y Cassiopeiae an atmosphere 
with an exponential outward decrease of density fits the observations. The density 
gradient has been found, and the density itself has been derived from the absolute 
intensities of the emissions integrated over the whole atmosphere. The result is 



log^ e = 11-8 



r 
0-74 — 



at the time of greatest extension of the gases. The total density is proportional to 
the electron density, because hydrogen and helium are almost completely ionized. 
The density decreases rather rapidly with the height, but nevertheless the effective 
atmosphere reaches to a radius of about lOr*. The emission chiefly originates in the 
outer parts, the absorption in an inner shell. Fig. 1 represents this model. 




Fig. 1. Be star model-atmosphere. p(r) is density of envelope, E(r) is contribution to line 

intensity (E > emission, E < absorption), (a) Photosphere, (6) limit of absorbing 

region, (c) limit of effective emitting region, (d) effective radius of absorbing layer, (e) 

effective radius of emitting shell 



P. Wellmann 311 

5. Ionization by Diluted Radiation 
In the same manner as in the construction of the curve of growth for absorption, the 
individual curves for different ions or elements will be shifted parallel to the 
log (NfH Jv)-a,xis and combined into one final function. According to (9) and (7) 
these shifts depend on the abundance and degree of ionization of the elements. 
For instance, it is possible to find N(Re u)/N(K n), N(Ke in) /N(Re n), and other 
ratios which demand an interpretation in terms of T s , T e , and W. 

The frequency of transitions between successive states of ionization, i and 
k = i + 1, may be expressed with the aid of the atomic absorption coefficient k 
as follows. 

The number of ionizations from level n of ion *, produced by radiation of frequency 
v, is 

dv 
±7rN i>n I v {T s ) . K n — cm-a-sec.- 1 ; 

the corresponding number of recombinations is : 

We now introduce the same assumptions concerning the departure from thermo- 
dynamic equilibrium as in the preceding sections, and the usual expression for the 
absorption by hydrogen-like atoms. After integration over all frequencies and 
summing up the contributions from all levels of the ion with the lower ionization, we 
have for the number of ionizations : 

Z a =W^K . N t |- e~ * . 2 ( -^-±^ . b n e& . <f> ns cm3-sec.-i. 

The analogous expression for the recombination is much more complex, therefore 
we restrict it to W = 1 and W <^ 1 : 

K w=i 

K 



877 2b k 1 h* 
ki ~ c* K ° • * e ' a> e B k ' (tomkT.)** 


Z - 3 e * ' 


where we write 




2 6 tt 4 we 10 f °° 1 dv 





(w) w < 1 

K(x) = -Ei(-x), 



B if B k are the generalized partition functions 



B = Ib n a> n e *t. , 
N { k is the number of all atoms in state i or k, Z is the atomic number, s the screening 
constant, and n the effective quantum number. 
Now the generalized equation of ionization is 



Zik — Z ki 



or, with W < 1, 



\4 X n 



N i+1 N e _ w B i+1 Z e (2nmkT e )*l> ^ _*. f * ' *™ 



N t b i+ltl B, h* 



f » 8 \hTJ 



312 The intensity of emission lines in stellar spectra 

It is easily seen that equations of the same kind given by previous authors are 
equivalent to (10) or special cases of this formula (see Stromgren, 1948). A numerical 
comparison with the earlier approximations (Wellmann, 1952a), for instance in 
the case of He, shows that an error amounting to a factor of about 10 can result from 
disregarding the singular population of the metastable levels and the recombination 
in the higher levels, especially if T is large. Therefore the use of (10) is unavoidable 
in some problems of emission line intensity. 

Most elements appear in more than two different states of ionization. We may ask 
whether it is permissible to calculate the abundances of higher ions by successive 
applications of (10). In principle, the degree of ionization is affected by transitions 
between all states of ionization, that is by the normal process as well as by the 
simultaneous escape or capture of several electrons. The probability of these 
multiple processes is very small, however. From a detailed discussion of the simplest 
case, the equilibrium between He i, He n, and He in, where the original Rosseland 
theory of three states applies exactly, it is found that the removal of the two electrons 
at a time has no appreciable influence. 

In the case of high ionization, for instance of Fe, the situation is not so clear, but 
it may be estimated that the contribution of multiple ionization only affects the 
number of ions when this number is too small to give observable lines. Therefore we 
can use (10) throughout the analysis of an emission spectrum. 

As an example, we may choose the spectrum of a nova. The intensities of H, He I, 
He ii, [O i], [O in], [Fe vn], and [Fe x] had been measured in the spectrum of 
Nova Serpentis 1948 in its early nebular stage, 14th-18th May, 1948 (Wellmann, 
1954). From the He-lines the abundance ratio He in/He n is deduced, using the 
calculated 6-values. The ratios O ni/O i and Fe x/Fe vn are found from more 
qualitative estimates of the dilution effects, which at least give the right order of 
magnitude. 

The observed intensities of the H, He-, and O-lines all refer to the whole volume 
of the main shell because they belong to the principal spectrum. The dilution factor 
is given by the geometrical relations according to (6). The radius of the main shell 
is the velocity of expansion multiplied by the time elapsed since the outburst, the 
radius of the star was about 1-0 r Q . Hence in the average W ~ 10 -9 . This equation of 
ionization contains W, N e , T s , and T e , but the latter is cancelled almost completely 
by the dependence of 6 on the electron temperature. The value N e = 10 7 is found from 
a model of the nova shell, which reproduces the observed spectrum-time relation. 
The ratio of He m/He n yields T s = 40,000° ± 3000°; the ratio H/He in yields 
T s = 46,000° ^ 3000°, assuming normal He-abundance. The same procedure with 
the [O in] and [O i] lines had the result: T s = 23,000° ± 2000°. T e is found from 
a relation between the intensity ratio [O in](N x + N 2 )/[0 m]/l4363, the ratio 
[Oi]AA6300, 6363/[0 i]A5577, the electron temperature, and the density; (Menzel, 
Aller, Hebb, 1941). It seems to be comparatively high, namely, T e = 18,000° 
± 5000°. The ratio Fe x/Fe vn, however, points to a much higher radiation tempera- 
ture, about 120,000°, if these ions were located in the main shell. But this discrepancy 
is partly removed if it is realized that the radius of the region containing ions of 
such high energy will be much smaller than the region of lower ionization (Wurm, 
Singer, 1953). In addition, this is supported by a discussion of the observed inten- 
sity ratio [Fe vn]/[0 in]. Hence W must be much larger than in the case of emissions 
originating in the main shell. The strength and the intensity ratio of Fe vn- and 



T. G. Cowling 



313 



Fe x-lines can be explained by assuming T s = 80,000° ± 4000° and, (in an 
atmosphere immediately surrounding the star), a density 10 -3 times that of the main 
shell. It should be noted that the resulting radiation temperature increases with the 
highest ionization potential involved, namely: On, 35 eV, 23,000°; Hen, 54 eV, 
40,000° ; Fe ix, 235 e V, 80,000°. 



References 

Ambartsumian, V 1933 

Bakes, J. G. and Menzel, D. H 1938 

Baker, J. G., Menzel, D. H. and Aller, L. H. . 1938 

Cillie, G. G 1936 

Goldberg, L 1941 

Menzel, D. H 1937 

1939 
Menzel, D. H., Aller, L. H. and Baker, J. G. . 1938 
Menzel, D. H., Aller, L. H. and Hebb, M. H. . 1941 

Menzel, D. H. and Baker, J. G 1937 

Rosseland, S 1926 

Stromgren, B 1948 

Struve, 1935a 

1935b 

Struve, O. and Wurm, K 1938 

Unsold, A 1938 

Wellmann, P 1952a 

1952b 
1952c 
1954 

Woolley, R. v. d. R 1947 

Wrxjbel, M. H 1949 

1950 

1954 

Wurm, K. and Singer, 1952 



Poulkovo Obs. Circ. No. 6. 

Ap. J., 88, 52. 

Ap. J., 88, 422. 

M.N., 96, 771. 

Ap. J., 93, 244. 

Ap. J., 85, 330. 

Popul. Astron., 47, 6 and 66. 

Ap. J., 88, 313. 

Ap. J., 93, 230. 

Ap. J., 86, 70. 

Ap. J., 63, 218. 

Ap. J., 108, 252. 

Ap. J., 81, 66. 

Ap. J., 82, 252. 

Ap. J., 88, 93. 

"Physik der Sternatmospharen", Berlin. 

Z. Astrophys., 30, 72. 

Z. Astrophys., 30, 88. 

Z. Astrophys., 30, 96. 

Phys. Verhandlungen, 5, 126. 

M.N., 107, 308. 

Ap. J., 109, 66. 

Ap. J., Ill, 157. 

Ap. J., 119, 51. 

Z. Astrophys., 30, 153. 



Dynamo Theories of Cosmic Magnetic Fields 

T. G. Cowling 

Department of Applied Mathematics, University, Leeds 
Summary 

The development of the dynamo theory of cosmic magnetic fields during the last twenty years is critically 
surveyed. After a general introduction to the theory, separate sections are devoted to the steady-state 
theory and the theory of turbulent magnetic fields. The problems raised but not answered by earlier 
work are generally surveyed, and the important questions needing a decisive answer are enumerated. 



1. Introduction 

In 1930, Professor Stratton was one of the examiners of my D.Phil. thesis at 
Oxford. In this thesis I discussed the dynamo theory, in the belief that it could 
provide the explanation of sunspot magnetic fields and possibly of the sun's general 
magnetic field. It therefore seems appropriate for me here to trace the development 
of the dynamo theory since then. 

The dynamo theory considered here is that which ascribes the maintenance of a 
cosmical magnetic field to currents induced in moving conducting fluid by the 



314 Dynamo theories of cosmic magnetic fields 

already existing field. The maintenance mechanism thus resembles that in a self- 
exciting dynamo. The recent development of such theories was initiated by Larmor 
(1919), in an attempt to explain sunspot fields and the sun's general field. He assumed 
the field, and the material flow, to be symmetric with respect to an axis, the lines of 
force and stream lines lying in planes through the axis. If material flows in towards 
the axis across the lines of force at great depths induced currents flow in circles 
round the axis here. Such currents are in the direction required to maintain the 
field, and Larmor suggested that they might in fact explain the origin of the field. 
If in the upper layers the material flows outward away from the axis across the lines 
of force, currents flow in the reverse sense here ; since, however, the material here is 
less conducting, the currents at greater depths can be expected to be dominant. A 
rough calculation suggests that the material motions necessary for maintenance of 
the field are very small — less than 10 -2 cm per sec for sunspot fields, and less than 
10 -6 cm per sec for the sun's general field. 

However, I was able in 1934 to show that a dynamo mechanism cannot work in 
the axially symmetric case. In this particular case the field possesses one or more 
"neutral rings". These are circles on which the field vanishes, with the axis of 
symmetry as axis ; near the rings the lines of force are closed loops threaded on to the 
rings. Since there is no magnetic field on a neutral ring, no current can be induced 
in the moving material at the ring ; moreover, since near the ring the material cannot 
move wholly inward (towards the ring) or wholly outward, the induced current 
cannot be in the same sense round the axis of symmetry at all points near the ring. 
But a field whose lines of force are loops threaded on the ring requires for its main- 
tenance a current which does not vanish on the ring, and which is directed in the 
same sense round the axis of symmetry at all points near the ring. It follows that in 
the axially symmetric case dynamo maintenance of the field is impossible. 

2. General Theory 

This result was regarded for some time as barring the possibility of dynamo 
maintenance of cosmic magnetic fields. The dynamo theory was indeed invoked again 
by Frenkel (1945) and by Gurevtch and Lebedinsky (1945), but their work 
appeared to have been done in ignorance of the result in question. However, 
repeated failures to provide an alternative mechanism led to a deeper examination of 
dynamo theories. My 1934 argument applied only to axially symmetric fields, or 
fields possessing a general similarity to these. The possibility that more general 
fields might be maintained by a dynamo mechanism now began to receive attention. 
The new attack on the problem was led by Elsasser (1946; 1947). He considered 
fields due to motions in a uniform incompressible sphere. Displacement currents 
can be neglected in the discussion ; thus if B is the magnetic induction, / the current - 
density, and fi the magnetic permeability 

curl B = 477/e/. .... (1) 

As a consequence of this div / = 0, so that any piling up of charge exerts no 
appreciable influence on the current flow. If E is the electric intensity and a the 
conductivity (both measured in E.M.U.), and v the material velocity, 

j=a(E + vx B) ....(2) 

and r- = curl E. (3) 

ot 



T. G. Cowling 315 

Eliminating / and E between these equations, and using the fact that div B = 0, 
we find 

— curl(v X B)\. (4) 

This equation indicates how the field varies in the presence of an assigned motion v. 
The motion is regarded as arbitrary, save that it satisfies the continuity equation 

div v = 0. (5) 

Elsassee, obtained a formal solution of equation (4) as follows. When v = 0, 
the equation indicates how a magnetic field decays as the energy of the currents 
flowing is converted into Joule heat. There are certain normal modes of decay in 
which the field decays exponentially; putting B = B r e~ x,t , the field in these normal 
modes is given, inside the sphere, by 

V 2 B r = — 4:7Tjua?i r B r . (6) 

The different possible fields B r have been considered by various authors, beginning 
with Lamb (1883). They form a complete set; that is, any field B generated by 
currents inside the sphere can be expressed in terms of a series 

B = 2KB r . ....(7) 

r 

Elsassee, assumed the field B satisfying (4), in the general case v ^ 0, to be ex- 
panded in a series of form (7), in which the coefficient b r are functions of the time. 
Equation (4) is then used to indicate how the coefficients vary with the time. 

The fields B r satisfy the orthogonality relation 

\-B r .Bflr=0 (r#«) ....(8) 

the integration being one over the whole of space. For if E r e~^ and j r e~ x,t are the 
electric intensity and current-density corresponding to B r , 

~\ir- /A = *" J E r • / A 

= \B r . curl I - B s ) dr 

= div I - B s X E r \ dr + curl E r . - B s dr. 

The first of these two integrals vanishes by Green's Theorem, since the tangential 
components of BJ/j, and E r are continuous at the surface of the sphere. Hence, using 
an equation similar to (3), 

— \ J r • /'A = K\ &r- B s dr. 

By symmetry, these also equal 

L \ - B„ . BAt. 



J IX 



316 Dynamo theories of cosmic magnetic fields 



Since r --£ s, equation (8) follows. The fields B r are also supposed to be normalized 
to satisfy 

B t Ht=\. ....(9) 



! 



In terms of j r , equations (8) and (9) are equivalent to 

= ^r {r = 8)m ....(io) 

Now by arguments similar to those used in deriving the orthogonality relations, 
if/, B, and B are the general vectors satisfying equations (2), (3) and (4), 

— I / . j r dr = 4tt J (E + V X B) . j r dr 

= -(-—. B r dr + 477 J (v X B) .j r dr. 

In this substitute the expansion (7) and a similar expansion for /*. Then simplifying 
by equations (8), (9), and (10), we get 



KK = - ^F + ^Ib s \(y x b.) .j r dr 



db 
Or -37= — KK + 2 a rs b s ( n ) 

dt s 
where a rs = 4tt J ( v X B s ) . j r dr. (12) 

Equation (11) indicates how the field given by (7) varies due to the joint action of 
electrical resistance and the motion. 

3. Stationary Fields 

The discussion now bifurcates. On the one hand, one may ask if steady motions 
of special kinds exist, able to maintain stationary fields ; on the other, the question 
is whether less regular motions may maintain fields which, though not precisely 
stationary, none the less possess certain statistically steady properties. We discuss 
first the possibility of the maintenance of stationary fields ; this has been considered 
especially by Bullard (1949). 

For a stationary field, equation (11) becomes 

AA = 2«rA- ....(13) 

s 

Putting r = 1, 2, 3, . . .we get an infinite number of equations to determine the 
infinite set of quantities b v b 2 , b 3 , . . . These equations in general possess only the 
trivial solution b x = b 2 = . . . = 0. However, a simple argument suggests that it 
should not be difficult to find motions for which the equations possess a non-zero 
solution, so that such motions can maintain the field corresponding to these values of 
b v 6 2 , . . . . 

The argument is as follows. Suppose that the motion v is increased in the ratio 



T. G. Cowling 317 

k : 1. Then each a rs is also increased in this ratio, so that the equations (13) are 
replaced by 

X r b r = k 2 a r J>s- ....(14) 

These equations can be solved to give non-zero values of b lt b 2 , . . . if k has one of 
a series of values, given formally by the determinantal equation 

ka 21 , ka 22 — X 2 , . . . =0. (15) 

Since the determinant on the left is an infinite one, the interpretation to be attached 
to the solution is as follows. Consider the set of equations (14), for r = 1, 2, . . . n, 
with the infinite sum with respect to s replaced by one in which s takes the values 
1,2,. . .n. These are n equations for the variables b lt 6 2 , . . . b n , which possess non- 
trivial solutions if a determinantal equation like (15) is satisfied, with n rows and 
columns in the determinant. Let k = k n be a root of this determinantal equation. 
As n -> oo, k n tends to a limit k' ; the ratios of the 6's simultaneously tend to limits. 
The solutions of equations (14) and (15) are given by these limits. 

This argument, however, appears to prove too much. In the axisymmetric case 
we have seen that no dynamo maintenance of a magnetic field is possible; but 
the above argument appears to indicate how a magnetic field capable of being main- 
tained by axisymmetric motion can be determined. The argument, of course, makes 
certain convergence assumptions, but this does not seem to be the cause of the 
contradiction. The cause appears rather to be that there is nothing to guarantee that 
the values of k are real. Equations (14) and (15) have a general resemblance with 
equations of eigenvalue theory. But in the latter equations the reality of k is guaran- 
teed by a symmetry condition a rs = a sr ; the quantities a rs defined by equation (12) 
satisfy no such symmetry condition. And a complex value of k is physically meaning- 
less in the present context. 

The spherical problem is difficult for detailed investigation, because the eigen- 
functions found by solving equation (6) involve Bessel functions. I have therefore 
considered a two-dimensional problem, in which the magnetic lines of force are 
wholly in planes z = constant, the field being independent of z. The conducting 
material is supposed to be confined between the planes x — a and x = — a ; the 
field, and the associated motions, are supposed periodic in y, with period 26. The 
motions and magnetic fields are supposed to be cellular, in the sense that the lines 
x = and y = rb (r integral) in any section z = const, are both lines of force and 
stream-lines of the motion ; the stream function, and vector potential of the field, 
are supposed to be reversed in sign if either x or y is reversed in sign. The lines of 
force and stream lines between y = and y = b are then not linked with those in 
regions outside these planes, and have a general resemblance with those inside and 
outside the sphere in Elsasser's problem. The volume integrals in equations (8), 
(9), (10), and (12) are replaced by surface integrals over the strip of a plane z = const, 
between y = and y = 6 ; otherwise the equations of section (3) apply unchanged 
to this slab problem. 

The argument of my 1934 paper establishes that, in the slab problem, no dynamo 
maintenance of a steady field is possible. In terms of equations (14) and (15), the 



318 Dynamo theories of cosmic magnetic fields 

explanation of this is as follows. In the slab problem it can readily be deduced from 
the equation defining a rs that a rs = — a sr . In consequence, all the roots k of equation 
(15) are either zero or pure imaginary, and so are physically meaningless. We can 
anticipate that the impossibility of dynamo maintenance in the axially symmetric 
problem in a sphere is equally due to the values of k becoming complex. 

These results cast doubts on the validity of the argument given in section (4). 
For dynamo maintenance of a steady field to be possible, it is not sufficient to estab- 
lish that equation (15) has a root k ; this root must also be shown to be real. Until 
it has been clearly proved that real roots exist for some special motions, the question 
whether a steady-state dynamo mechanism exists must be left open. A numerical 
method based on successive approximations, like that sketched at the end of section 
4, is peculiarly ill-fitted to supply such a proof. For even if the solution of the nth 
order determinantal equation approximating to (15) is real, there is no guarantee 
that the corresponding root of the (n + l)th order equation will not be complex. 
It would be peculiarly unsatisfactory to go through the laborious numerical work 
simply to satisfy oneself that a particular motion cannot maintain fields by dynamo 
action. Clearly some basic physical idea must be sought why, if at all, some motions 
may exist which maintain steady fields. 

However, these analytical difficulties do not provide the only reason why the 
steady-state dynamo theory is less popular than it might be. If the theory is 
correct, for motion of a given pattern a steady field is maintained only if the motions 
have a given strength (corresponding to a given k satisfying (15)). One may ask what 
happens if the motion exceeds this strength. At first sight the answer would appear 
to be that in this case the field is amplified, and steadily grows. But this is not 
necessarily correct, as the following argument shows. 

Equation (4) indicates that BB/dt is the sum of parts d x Bjdt, d 2 B/?t, where 

d,B 

-±— = curl (v X B) (16) 

¥-£™ ■••■<»> 

The part d x BJdt represents the variation of B which would occur if the conductivity 
a were infinite. In this case the lines of force can be regarded as "frozen" into the 
moving material ; the changes in the field are due to the transport of lines of force 
from point to point with the material. The effect of such transport is in general to 
stretch the lines of force and (because the quantity of matter in a given tube of force 
remains constant) consequently to press the lines of force together laterally, so 
increasing the field-strength. Equation (17), on the other hand, gives the variation 
in B which would occur in the absence of mass motions, due to the decay of the 
currents maintaining the field as their energy is converted into Joule heat. During 
such decay, the lines of force tend to shrink up and disappear. A steady magnetic 
field can be maintained only if the extension of lines of force due to the mass motion 
is exactly balanced by the shrinkage due to the decay process. 

Suppose that a motion maintaining a steady field has been found ; suppose then 
that the motion is increased in strength at every point. The effect of the increase is 
that the stretching of the lines of force is no longer balanced by the shrinkage due to 
the decay ; the lines of force are stretched, on balance, and their stretching goes on 



T. G. Cowling 319 

indefinitely. This means that the field grows, but not necessarily that it grows 
regularly . It may simply grow more and more irregular, lines of force in one direction 
being continually brought into close juxtaposition with lines of force in other, and 
perhaps opposed, directions. For the steady-state dynamo theory to be worth while, 
the field must be "stable" against such increases in the motion; that is, such 
increases must produce increases in the field which preserve certain regular character- 
istics until the field has grown so large that it is able to react on and limit the motion 
by its mechanical effects. Thus it is not sufficient only to know that a given motion 
maintains a steady field ; when increased, it must not at once destroy the regular 
features of this field. 

Such conditions are, of course, satisfied for the dynamos in common use. Bondi 
and Gold (1950) have suggested that the reason why these dynamos work is that the 
currents flow in a multiply-connected volume (a closed circuit). They have con- 
sidered dynamo effects in the case of infinite conductivity, which is equivalent in 
many respects to assuming finite conductivity and infinitely great motions. They 
found that, in this case, the external field cannot increase indefinitely unless the 
currents flow in a multiply-connected space. Their results depended on the fact that 
in perfectly conducting material no new lines of force can leak across the surface of 
the material, and to that extent are unrepresentative of the results for finite conduc- 
tivity. But they are important as indicating that in a simply-connected region one 
does not get an indefinite regular amplification of the field simply by increasing the 
dynamo motions indefinitely. Rapid motion inside a conducting sphere does not 
lead to a correspondingly rapid increase in the external field ; it leads to rapid changes 
in the internal field, but since the only rapid changes in the external field with which 
these can be correlated are irregular ones, any increased internal field must itself be 
irregular. 

4. Turbulent Fields 

The dynamo maintenance of non-steady fields has usually been considered in con- 
nection with turbulent motion. Since moving material tends to drag the lines of 
force with it, turbulent motion tends to twist and stretch the lines of force until 
these are distorted into an irregular tangle. However, just as turbulent motion 
possesses certain statistically steady features, the resulting turbulent magnetic field 
can also be expected to possess statistically steady properties. It is, moreover, 
suggested that from time to time the external field may possess a fairly appreciable 
dipole component. 

However, the suggestion most frequently made (Fermi, 1949) is that a steady 
state tends to be set up, in which there is equipartition of energy between the 
turbulent motion and the turbulent magnetic field, for all "wavelengths" of the 
turbulence "spectrum" of each. Equipartition is supposed to be set up most rapidly 
in the short wavelengths, which contribute only a small fraction of the energy, and 
to extend later to progressively longer and longer wavelengths. Equipartition cannot 
be expected save between quantities which behave similarly and interchange energy 
on equal terms. Elsasser (1950) has put forward the following argument to suggest 
that material motions and magnetic forces do, in fact, behave very similarly. 

Using the equations div B = 0, div v = 0, equation (4) can be put in the form 

^ + (v . V)B = (B . V)v + — *- V*B. .... (18) 



320 Dynamo theories of cosmic magnetic fields 

The equation of motion of the material is 

P \-^+ (v.V)v) = -Vp + F+ K V*v + j x B ....(19) 

where p is the density, p the pressure, F the body force, and k the viscosity. Put 
v + B/V^w) = P, v - BJy/{±iw) = Q ) 



K i = i K + p/(87r^a), ic 2 = |k — pK&irpo). 

Then (18) and (19) can be combined to give 



(20) 



9 (l£ + iQ • V)P ) = ~ V i P + 8^) + F + " lV * P + K ^ Q ■ ■ * ' (21) 
p(^ + (P. V)0) = - V (^ + ^)+ F + ^ V2 <? + ^V 2 P (22) 

Remembering that B 2 /8ttju is the magnetic "pressure", these equations are seen to 
be closely analogous with the form taken by the equation of motion (19) in the 
absence of a magnetic field, the variables P and Q corresponding to the velocity v. 
Remembering the definitions of P and Q, it follows that B\^{4^pyi) should fill a 
role closely analogous to that of v. This suggests that the magnetic energy B^/Hvju 
per unit volume fills a place essentially like that of the material energy |pv 2 . 

Batchelor (1950) and Chandkasekhae, (1950), combating this., point of view, 
point out that equation (18), satisfied by B, is identical in form with the equation 
satisfied by the vorticity in the motion of a viscous liquid, \\kn(jLa corresponding to 
the kinematical viscosity. They therefore suggest that the correct analogy for B is 
with the vorticity. In turbulent motion, as is well-known, the vorticity becomes 
progressively more important, compared with the velocity, as one passes to smaller 
and smaller wavelengths in the turbulence spectrum. Thus if B behaves like the 
vorticity equipartition is unlikely for all wavelengths ; it may prevail for very small 
wavelengths, but for greater the magnetic energy is likely to be the smaller. 

The defect of this argument is that the vorticity is functionally connected with the 
velocity; the vector B is not. Moreover, it rests only on equation (18). Equation 
(19), indicating how mechanical forces of magnetic origin affect the motion, is not 
used, and it is through this equation that equipartition might be expected to be set 
up. The main importance of the argument is in indicating how dangerous it is to build 
simply on analogy. Equipartition of energy is not proved by any analogy ; it is 
proved only by actually establishing that magnetic energy and material kinetic 
energy interact with each other on equal terms. 

One obvious difference exists between magnetic energy and material kinetic 
energy. Turbulent motion can be generated and maintained by large-scale shearing 
motion ; a large-scale magnetic field cannot, by itself, generate a turbulent magnetic 
field. Thus some limitations of equipartition must exist ; how serious these limitations 
can be demands closer investigation. 



T. G. Cowling 321 

The equation of motion in the absence of a magnetic field is 

-^ = - ( V . V)v + - (F - Wp + /cW). 
at p 

In discussing the application of this equation to turbulence, the following argument 
is often used. If the turbulent motion is divided into an energy spectrum, the term 
— (v . V)v in the equation involves the products of different spectrum elements. 
These products contribute to the spectrum of 3 v/dt interaction terms which may have 
longer wavelengths than the elements which compose the products, but more often 
have shorter wavelengths. Thus there is a continuous tendency for large eddies to 
break up into smaller and smaller eddies. Energy is continually being fed into large 
eddies from the kinetic energy of mass motions ; it passes from these to progressively 
smaller and smaller eddies, and is destroyed by viscosity when it reaches the smallest 
eddies. 

A similar argument can be applied to equation (18). The terms (v . V)B, (B . V)v 
contribute to the spectrum of 3B/dt interaction terms which more often than not 
have shorter wavelengths than the elements of B and v which interact. In ordinary 
language this means that turbulent motion tends to twist existing lines of force 
into an increasingly tangled confusion, rather than to generate large-scale fields from 
small-scale ones. Thus, just as in turbulent motion small eddies are supplied by the 
interaction of large eddies, so in turbulent magnetic fields small-scale elements of the 
field are supplied by the interaction of larger elements with the eddies. If the energy 
of large-scale elements of the field is small compared with the kinetic energy of large 
eddies, the energy of small-scale elements of the field must equally be smaller than 
the kinetic energy of small eddies. 

If this argument is sound, it implies that there is no reason to expect equipartition 
of energy between turbulent magnetic fields and turbulent motion, unless the condi- 
tions for equipartition have already been met by imposing a large-scale field whose 
energy equals that of the large-scale motion. There is, of course, no automatic 
reason why equipartition of energy should obtain in any particular problem. In the 
kinetic theory of gases it obtains between molecules of different gases in a mixture 
because these can interchange energy on equal terms, without loss, at collision. In 
turbulence, on the other hand, there is no "equilibrium" state ; energy is continuously 
being passed from long wavelengths to shorter. Elements of the magnetic field do 
not interchange energy simply with turbulence elements of the same size, and so 
there is no obvious reason to expect equipartition between the two. 

Only in one case is equipartition known to occur; this is the case of Alfven's 
magnetohydrodynamic waves, where there is equipartition between material kinetic 
energy and the energy of the disturbance field b due to the wave. In this case equi- 
partition is to be expected on general grounds ; dissipation of energy is ignored, and 
the second-order terms like (v . V)v in (19), or (v . V)& in (18), cancel out. Thus 
the whole process of degradation of energy into shorter wavelengths is stopped ; the 
only interaction terms in the equations are those due to interaction of v or b with the 
undisturbed field B . Since B is assumed to be uniform, these last terms introduce 
into dvjdt and abfat only those wavelengths already present. Hence in this case 
kinetic energy is freely interchanged with magnetic energy of similar wavelengths 
and equipartition follows. However, the factors securing equipartition in this 
instance are not met in more general circumstances. 



322 Dynamo theories of cosmic magnetic fields 

The advocates of equipartition can easily criticize the above arguments, since, 
like the arguments in support of equipartition, they rest on analogy rather than 
detailed analysis. I personally find them more convincing than the arguments in 
favour of equipartition. But clearly any conclusion must remain doubtful until 
confirmed by detailed analysis. 

5. Conclusions 

To sum up, the steady-state theory is under somewhat of a cloud at present. A 
proof is needed of the possibility of dynamo maintenance of steady fields by 
currents in a simply-connected region. When this has been given, a further proof 
must be given that the field is not twisted out of all recognition if the dynamo motions 
are slightly increased. Until such proofs are given, the validity of steady-state 
dynamo theories must be regarded as open to question, and the proofs appear to 
require some altogether new idea. 

The non-steady dynamo theory is in a much less developed state. Not even has 
it been definitely proved that a sufficiently violent turbulent motion can spon- 
taneously generate a turbulent magnetic field from nothing, and any more ambitious 
assertion is open to criticism. Nevertheless, the possibilities open if non-steady 
dynamo mechanisms work are so interesting in many directions that any positive 
results in this field will be very worth while. 



Refeebnces 

Batchelor, G. K. . . 1950 Proc. Roy. Soc, 201, 405. 

Bondi, H. and Golo, T. . . . . 1950 M.N., 110, 607. 

Btjllard, E. C 1949 Proc. Roy. Soc, 197, 433. 

Chandrasekhar, S 1950 Proc. Roy. Soc, 204, 435. 

Cowling, T. G 1934 M.N., 94, 39. 

Elsasser, W. M 1946 Phys. Rev., 69, 106; 70,202. 

1947 Phys. Rev., 72, 821. 

1950 Phys. Rev., 79, 183. 

Fermi, E 1949 Phys. Rev., 75, 1169. 

Frenkel, J 1945 C.R. Acad. Sci. U.S.S.R., 49, 98. 

Gtjrevich, L. E. and Lebedinsky, A. I. . 1945 C.R. Acad. Sci. U.S.S.R., 49, 92. 

Lamb, H 1883 Phil. Trans., 174, 519. 

Larmor, Sir J 1919 Brit. Assoc. Reports, p. 159. 

1934 M.N., 94, 469. 



On the Interpretation of Stellar Magnetic Fields 

S. K. Runcorn 

Gonville and Caius College and Department of Geodesy and 
Geophysics, Cambridge University 

Summaby 

The present knowledge of stellar magnetic fields is reviewed. Their large variation with time in many 
stars presents a difficulty, the explanation of which in terms of mechanical oscillations is not satisfactory. 
In this paper a new attempt is made to examine under what conditions the measurements can be recon- 
ciled with the assumption of a steady field. This "oblique rotator" theory can only be true if the fields 
contain strong multipole components and a suggestion is made as to why this is probable. In addition 
certain elements must be concentrated in certain areas of the stellar surface to account for the intensity 
and Doppler displacement variation observed for certain spectral lines. 



1. Introduction 

The explanation of the magnetic fields associated with the Earth, the Sun, and 
certain stars seems likely to be found in the flow of electric current in the interior of 
these bodies. At various times it has been held that these fields were a new property 
of massive bodies in rotation but experimental tests of certain simple consequences 
of such "fundamental theories" have latterly proved this yiew to be unfounded. 
Runcorn, Benson, Moore, and Griffiths (1951) have measured in deep mines the 
radial gradient of the geomagnetic field and they find that the magnetic field just 
inside the Earth is irrotational, which it would not be if matter acted as a virtual 
current, i.e. a source of magnetic field. Also Blackett (1952) has proved by a 
laboratory experiment the non-existence of magnetic fields near dense bodies of the 
magnitude demanded by this hypothesis. The original objection felt by Schuster 
(1912) to explanations of the magnetic fields of cosmical bodies in terms of current 
flow was that the maintenance of an axial magnetic field demands electromotive 
forces directed round the fines of latitude, the existence of which in a spherically 
symmetrical body does not admit of any ready explanation. Now it had long been 
known that inside a stationary electrically conducting sphere magnetic fields may 
exist of "toroidal" type which do not appear outside the conductor. Elsasser 
(1947) raised the possibility that such fields exist within the Earth's core, thus, in 
effect, allowing the simple axial symmetry of the magnetic field, which led to the 
difficulty mentioned above, to be removed. Bullard (1948) and Runcorn (1954) 
have shown how such fields might be excited by simple physical processes in the 
Earth's interior. 

Further progress in the understanding of cosmical magnetic fields would seem to 
depend on the correct appraisal of the evidence now available of the apparent 
variability of the fields of the Earth, the Sun, and the early-type stars. The gradual 
recognition of the short time scale, geophysically speaking, of the geomagnetic 
secular variation has enabled inferences concerning the fluid motions in the Earth's 
core to be made, which even now can be seen to be quite fundamental to the explana- 
tion of the main field itself. There seems little doubt, for instance, that the irregular 

323 



324 On the interpretation of stellar magnetic fields 

fluctuations in the length of day, the proof of the existence of which forms a fasci- 
nating chapter of astronomical literature, have their origin in the same motions of 
the Earth's core, which are a key factor in the understanding of the generation of 
the geomagnetic field. In time inferences drawn from the astronomical data will add 
considerably to our knowledge of the physical properties of the interior of the Earth. 
Further, there is now strong evidence, notably that of Hospers (1953, 1954) on the 
remanent magnetization of the lavaflows in Iceland, that the geomagnetic field has 
actually reversed several times since early in the Tertiary epoch. If this interpreta- 
tion of the reversed polarization of igneous rocks is finally maintained it may well 
prove of decisive importance in choosing between the various theories of the cause 
of the electromotive forces in the Earth's core. Hale's original measures of the 
Sun's field as a dipolar one with a polar magnitude of 50 gauss seem, on internal 
evidence of a convincing type (von Klttber, 1952), to be not mistaken. Yet various 
observers find a value not greater than about 5 gauss over the last decade. Theoretical 
progress is somewhat hampered by doubt as to the reality of these considerable 
variations of the terrestrial and solar magnetic fields. 

But perhaps the behaviour of the magnetic fields of the early-type stars detected 
in pioneer work by H. W. Babcock, presents the most considerable and, from the 
point of view of theory, most urgent problem of interpretation. Babcock (1953) 
finds among stars later than type B8 fourteen in which the magnetic fields actually 
reverse, ten in which the fields do not reverse but show large fluctuations in magnitude 
and eleven in which the fields are steady. Many of the magnetic variables are 
apparently spectrum variables, in which certain fines undergo fluctuations of intensity 
of the same period as the magnetic field, the lines usually belonging to two groups 
showing oscillations in antiphase. The stars with steady fields likewise show no evi- 
dence of variability in spectrum. Two of the stars in which the measured fields reverse 
have been investigated in detail; HD 125248 by Babcock (1951) and a 2 Canum Vena- 
ticorum by Babcock and Burd (1952). Fluctuations in the velocities in the line of 
sight for the various elements are found with periods equal to that of the magnetic 
field, though the velocity curves are not sinusoidal, second harmonics being notably 
present in the velocities of the Cr n and Ti n lines of a 2 Canum Venaticorum. A 
further notable feature is that in the star HD 125248 the intensities and also the 
velocities in the line of sight of the rare earth elements (e.g. Eu n) and the chromium 
lines (Cr n) oscillate in antiphase. The velocity differences observed in the lines 
of these stars are of the order of a few kilometres per second, representing move- 
ments of the elements of the order of the stellar diameter during the period. It is, of 
course, most essential that a satisfactory physical model of these magnetic variables 
should be found and in particular to decide whether these reversals are to be taken as 
wholesale alterations of the distribution of the magnetic lines of force. 

On the latter view it seemed impossible, in view of the high electrical conductivity 
of stellar material, that the magnetic fields at a point in the star could actually 
change with respect to an element of the atmosphere in such short times and this 
consideration prompted the suggestion of Runcorn (1948) that the form of the 
magnetic field was being periodically altered by mechanical oscillations of the star. 
Using the well-known device, appropriate for materials of high conductivity, of con- 
sidering the lines of magnetic force attached to particles of matter, so that motion of 
the fluid may stretch lines of force but not create new ones, it can be shown that 
motions in the star may easily redistribute the lines of force over the surface. That 



S. K. Runcorn 325 

this might lead under certain circumstances to an apparently reversing field is due 
to the fact that, in the interpretation of a measured Zeeman Shift, Babcock assumes 
an axial dipolar magnetic field and coincidence between the axis of rotation and the 
line of sight. Now the magnetic field actually measured is the mean component in 
the line of sight of the field integrated over the visible hemisphere, allowing for limb 
darkening, and on the above assumptions is the difference between the (larger) con- 
tribution from the polar regions and that from the equatorial regions. It was 
suggested that the effect of a radial pulsation of the star would be to sweep the 
lines of force alternatively towards and away from the poles, when the star is expanding 
and contracting respectively, thus making the polar and equatorial contributions 
alternatively dominant. In the papers of Schwarzschild (1952), Ferraro and 
Memory (1952), Gjellstad (1952), and Cowling (1952) are to be found detailed 
mathematical consideration of the "magnetic oscillator" theory. These discussions 
are not entirely favourable to it for Schwarzschild's suggestion that the oscillation 
is a free one under its own magnetic field is shown by Ferraro and Memory to imply 
impossibly large magnetic fields in the interior of the star and though Cowling shows 
that gravity oscillations are more likely to occur and could lead to sufficiently long 
periods, if the motions had very strong horizontal components, he is able to demon- 
strate that the resulting redistribution of the lines of magnetic force is not of the type 
required to lead to apparent reversals in the field. 

2. Oblique Rotator Theory of Magnetic Variables 
It is consequently of interest to reconsider the alternative explanation; termed 
the oblique rotator theory, discussed by Babcock (1951), on the whole unfavourably 
at that time. This assumes that the field of the star is steady but that due to the 
rotation of the star the field presented to the observer changes with time. It is easily 
seen that for this to happen it is necessary for the axis of rotation of the star, the 
line of sight, and the magnetic axis to have different directions in space. Suppose 
first that the magnetic axis and the line of sight are inclined at angles a and ft respec- 
tively to the axis of rotation. If the surface field of the star is dipolar in character 
then Schwarzschild (1950) has shown that the effective field strength H e , as 
measured by the observer, including the effect of limb darkening given by the 
coefficient u, is 

„ 15 + u H v 

H e = . — - cos i 

lo — 5u 4 

where H p is the field strength at the magnetic pole and i the inclination of the 
magnetic axis to the line of sight. 

Now for the oblique rotator by the use of a simple formula of spherical trigo- 
nometry we obtain 

cos i — cos a cos ft + sin a sin ft cos 2rd\T , 

where T is the period of rotation and t is the time measured from the moment when 
the three axes are coplanar. 
Then 

H e = 0-303^(008 a cos ft + sin a sin ft cos 2tt*/T), 

taking u — 0-45, the value given by Babcock (1951), as typical of AO stars. 



326 



On the interpretation of stellar magnetic fields 



This demonstrates an important apparent objection to the oblique rotator theory, 
rather similar to one raised by Babcock (1951), i.e. there is in the curves of H e 
against t for the magnetic variables a distinct second harmonic. For instance, for 
the star H.D. 125248 Babcock (1951) finds 

H p = + 2000 + 6600 cos 2ift/T — 1600 cos 4nt/T (gauss) 

and for the star a 2 Canum Venaticorum from the mean curve given by Babcock and 
Btjrd (1952) the expression for the apparent polar field is 

H v = — 1100 — 3800 cos 27rt/T + 1800 cos 4*rt/T (gauss) 

where t is measured in each case from the Eu n maximum. 

Thus assuming the axis of rotation is fixed in space, the oblique rotator theory can 
only be maintained by assuming that the surface magnetic fields contain higher har- 
monics. However, it seems for mechanical reasons to be difficult to justify the exis- 
tence of a purely dipolar field. For the production of an axial dipole field outside a 
conducting sphere it is necessary to have a distribution of electric current flow over 
concentric spherical surfaces directed around lines of latitude, the intensity of flow 
per unit distance measured along lines of longitude being proportional to the sine of 



illNE OF SIGHT 



MAGNETIC AXIS 




Fig. 1. The oblique rotator: diagram of notations 



the angle of colatitude (0). Particularly if it is assumed that the current flow pro- 
ducing the observed field is concentrated in the outer layers of the star, considerable 
mechanical forces would be necessary to preserve this distribution against the 
resulting electromagnetic forces. Suppose that the magnetic field is produced by a 
surface distribution of current (I sin 6). The radial component of the resultant mag- 
netic field will be continuous through the current sheet and will be given by H r 
= (87t/3)J cos 6. Thus the electromagnetic force on the atmosphere per cm 2 at 
colatitude 6 will be of magnitude (3/8ir)H 2 r tan d and will be directed towards the 
equator. In the case of the stars this force will clearly be of the order of 10 8 dynes. If 



S. K. RtiNCOEN 327 

the currents are distributed only over the whole depth of the atmosphere, say 100 km, 
the electromagnetic forces per cubic centimetre of gas will be 10 dynes. Now the 
gravitational forces in a stellar atmosphere are of the order of 10 -6 dynes per c.c, so 
that there is a strong tendency for the current to be concentrated in the equatorial 
regions, i.e. the state of minimum magnetic energy. This enhanced equatorial current 
would, of course, give rise to a strong axial octupole component in the surface mag- 
netic field and on the oblique rotator theory result in a magnetic field variation 
markedly different from a purely sinusoidal one. 

The potential <f> of an axial octupole in spherical co-ordinates is given by 



* = ^(") 4 p,(«»n 



where H p is the polar field. This may be converted to spherical co-ordinates about 
the line of sight by the addition theorem of spherical harmonics. The terms for 
m^Odo not contribute to the integral and are therefore omitted. Thus 



<f> = -~~ {-) -P 3 (cos 0)P 3 (cos i) + • 



Converting to cylindrical co-ordinates about the line of sight, the component of the 
field in the line of sight may easily be found by differentiation of <f> with respect to z. 
This is found to be 

H ( \ (-8a*-35p4 + 40ay) 

H *\P) = g^ • H v . P 3 (cos »). 

The field measured by the observer is then 

H e = f" 2tt P . H Z ( P ) . (1 - u + uVa? — p 2 /a)d P 

Jp=0 

1 + lu H v 



3—u 



P 3 (cos *) 



The apparent field will fluctuate with second and third harmonic components of the 
period of rotation. 

An important study by Deutsch (1952a) of the spectrum variable H.D. 124224, 
which has a period of 0-52 days, rather strongly suggests that spectrum variability 
involves a distribution of the sources of certain absorption lines with longitude with 
at the same time a rotation about an axis inclined to the line of sight. In a general 
discussion of twelve spectrum* variables of type A, Deutsch (1952b) is able to show 
that the differences in profile of a certain line (Mg n 4481), common to all these stars 
and of constant intensity and therefore presumably uniformly distributed over the 
surface, are predictable from estimates of the peripheral velocities of the individual 
stars as deduced from their period of variability. This evidence, together with the 
direct verification discussed elsewhere in this book* that the sources of certain 
spectral lines can be concentrated in certain areas of the stellar surface, provides 
strong indication that the oblique rotator theory of magnetic variables must be 
reconsidered, as Deutsch himself suggests (1952b). 

* See Section 14 (Spectral Variations and Novae), Volume 2. 



328 On the interpretation of stellar magnetic fields 

3. Spectrum Variability 

To make a first attempt to discuss quantitatively the differences observed at 
different phases in the intensities and Doppler shifts of certain of the elements, as, 
for example, in H.D. 125248, we note that the velocity in the line of sight at a point 
on the stellar surface arises from the component of the angular velocity (2tt/T) 
perpendicular to the line of sight. Thus, taking the line of sight as the axis of spherical 
co-ordinates, the velocity in the line of sight at colatitude 6 and azimuth X, measured 
from the plane containing the line of sight and axis of rotation, equals 
(277-/T) sin /? sin 8 sin X, taking the radius of the star as unity. We will further assume 
that the sources of certain lines are not distributed evenly over the stellar surface. 
Deutsch (1952b) has suggested that the magnetic field affects the excitation by 
inhibiting convection and so disturbing the temperature gradient. It thus seems 
generally reasonable to take a distribution of sources given by a zonal harmonic with 
respect to the magnetic axis. In the model of the magnetic field distribution, which 
we considered earlier, the magnetic field is mainly vertical at the poles and horizontal 
at the equator, thus it seems reasonable to represent the source distribution as a sum 
of even zonal harmonics, say, a -f a 2 P 2 (cos y), where y is the angle of colatitude 
with respect to the magnetic axis and a and a 2 are constants with a greater than 
a 2 /2. We convert this to a sum of tesseral harmonics along the line of sight as the 
axis of co-ordinates. Then by the additional formula of spherical harmonics 

2 (2 — m)\ 
P 2 (cos y) = P 2 (cos i)P 2 (cos d) + 2 £ )— ~ P£(cos i)P™ (cos 0) cos m{X— X), 

ot==1 \2 T m )- 

where X' is the azimuth of the magnetic axis at any time t, given by the expression 

sin X' = (sin a . sin 277-£/T)/sin i. 

By integration over the visible hemisphere the mean intensity (I) of lines which 
are distributed in the above way over the stellar surface and the mean velocity in the 
line of sight (v) as given by the Doppler shift may be derived. The surface brightness 
is taken as proportional to the expression (1 — u + u cos 6), 

I = (1 — u -f u cos d)(a + a 2 P 2 (y)) cos 6 sin OdOdX 

J/i=o Je=o 

= (1 — u/3)a + 1/4(1 + w/15)a (3 cos 2t - 1) 

= (3 — u)aJ3 + (15 + u)a 2 /G0 + (15 + ^)a 2 (2 cos 2 a cos 2 /? + sin 2 a sin 2 /J)/40 

15 _i_ u 
-I — — . a 9 . sin 2a sin 28[cos 2-nt IT +1/4 tan a tan ft cos ^t/T]. 

As a 2 cannot exceed 2a the greatest fluctuation in the light from this harmonic cannot 
exceed 2-4:1, if w = 0-5. This is perhaps rather less than the fluctuation observed and 
indicates that in a fuller discussion it might be necessary to include higher harmonics. 

/*A=27T / > 6=77/2 

v = (l//) (27ra 2 /3T) sin £P|(cos i)P 2 (cos 6) sin X'(l —u + ucosd) 

cos 6 sin 2 6 sin 2 X dfldA 
= (1/7) . (1/5 — 3w/40)(7r 2 a 2 /T) sin 2a sin 2$>in 27rt/T + 1/2 tan a tan sin 4vrt/T]. 

A detailed comparison of these results with the observational data will not be 
attempted here but an inspection of these expressions for I and v show up two points 



S. K. Runcorn 329 

of interest. Firstly, the line intensity at certain phases varies rather rapidly due to 
the presence of the second harmonic : this is in accord with Deutsch's observations. 
Secondly, v may vary rapidly when / is small : the rapid acceleration of the Eu n 
lines while strong just before the phase of maximum magnetic intensity is a notable 
feature of Babcock's results on H.D. 125248. It is clear also that the oblique rotator 
theory provides an explanation for the light curve and for the cross-over effect found 
in H.D. 125248, from which Babcock has inferred that different areas of the star 
have different polarities and radial velocities. The theory also accounts in a general 
way for the systematic difference in Zeeman shift measured in the constant and 
varying lines. 

Babcock (1951) objected further to the oblique rotator theory on the grounds that 
the antiphase fluctuations in intensity of the rare earth and chromium group of 
lines require each group to be concentrated at one magnetic pole and he points out 
the real difficulty of thinking of a process of separation which would take account of 
the polarity rather than the intensity of the field or its inclination to the stellar 
surface. Further, in H.D. 125248 and a CVn the rare earth lines and Cr lines vary 
in antiphase, but the relationship of magnetic polarity to the respective elements is 
opposite in the two stars. The above calculation of the line intensity shows that a 
source distribution can be used which is not open to this objection. 

The FO type star, y Equilei, found to have an unvarying magnetic field by 
Babcock (1948), has extremely fine metallic lines and so presumably rotates about 
an axis inclined little to the line of sight. There is, of course, no evidence as to whether 
its magnetic axis is inclined appreciably to the line of sight. 

4. Conclusion 

Thus it seems possible now to explain the apparent variability of the magnetic 
fields of stars in terms of the various aspects presented by the magnetic and rotational 
axes to the observer. This view might to some extent have been inferred from the 
fact that of the stars in which magnetic fields have been detected, some have measured 
fields which are constant, some which are variable and some which actually reverse. 

Little progress has, of course, been made towards the understanding of the origin 
of the stellar magnetic fields, but it is not out of the question in view of the long free 
decay time that these fields are residual fields and have their origin in some event in 
stellar evolution. The reason for the non-axial character of the fields we have inferred 
above is also obscure. But there seems no reason for the coincidence of the magnetic 
and rotational axes as there is in the case of the Earth, see Runcorn (1954). 

The understanding of cosmical magnetic fields is a considerable challenge to the 
extra-laboratory sciences. It is clear that in both the geophysical and astrophysical 
examples we are still at the stage of providing a reasonable physical model to take 
account of all the facts available. Satisfactory theories as to the cause of the electric 
current flow postulated to explain the fields must probably await the completion 
of this task. 



References 

Babcock, H. W 1948 Ap. J., 108, 191. 

1951 Ap. J., 114, 1. 

1953 M.N., 113, 357. 

Babcock, H. W. and Bxjrd, S 1952 Ap. J., 116, 8. 

Blackett, P. M. S 1952 Phil. Trans., A 245, 309. 



330 Theories of variable stellar magnetic fields 

Cowling, T. G 1952 M.N., 112, 527. 

Deutsch, A. J 1952a Ap. J., 116, 536. 

1952b Proc. Joint Comm. on Spectroscopy, 
I.A.U. Trans., vol. VIII. 

Elsasser, W. M 1947 Phys. Rev., 72, 821. 

Ferraro, V. C. A. and Memory, D. J. ... 1952 M.N., 112, 361. 

Gjellestad, G 1952 Ann. Astrophys., 12, 148. 

Hospers, J 1953 Proc. K.Ned.Akad. Wetensch.,B, 56, 467. 

1954 Proc. K. Ned. Akad. Wetensch., B, 57, 112. 

Kxuber, H. von 1952 M.N., 112, 540. 

Runcorn, S. K 1948 Trans. Oslo Meeting, Assoc, of Terr. Mag, 

and Elect., I.A.T.M.E. Bulletin No. 13, 
p. 421. 
1954 Trans. Amer. Oeophys. Un., 35, 49. 

Runcorn, S. K., Benson, A. C, Moore, A. F. 

and Griffiths, D. H 1951 Phil. Trans., A 244, 113. 

Schuster, A 1912 Proc. Phys. Soc. (London), 24, 121. 

Schwarzschild, M 1950 Ap. J., 112, 222. 

1952 Ann. Astrophys., 12, 148. 



Theories of Variable Stellar Magnetic Fields 

V. C. A. Ferraro 

Queen Mary College, University of London 



(1) As early as 1892 Schuster conjectured that massive celestial bodies in rotation 
might, like the Earth, be the seat of magnetic fields (see also Schuster, 1912). It is 
said that one of the reasons which led Hale to consider the project of the 100-in. 
telescope at Mount Wilson was the hope that it might reveal the existence of stellar 
magnetic fields. Hale (1908) was successful in establishing the existence of sunspot 
magnetic fields, but the discovery of stellar magnetic fields was reserved for his 
compatriot, H. W. Babcock (1947), who discovered that the star 78 Virginis 
possessed a general magnetic field of the order of 1500 gauss at the pole. 

Since then Babcock has discovered that several other stars possess magnetic 
fields exceeding 1000 gauss; furthermore, he found that in some the magnetic field 
was variable, whilst in others, notably HD 125248, the field was not only variable but, 
what is remarkable, its polarity reversed once every 9j days. 

All magnetic stars appear to be spectral variables of the types A and F in rapid 
rotation. The variability of the star HD 125248 has been the subject of a careful 
study by Babcock (1950) who found that the polar magnetic field varies from about 
7000 gauss to — 6000 gauss. Radial velocity measurements show an harmonic 
variation which seems coupled with changes in the magnetic field, the velocity ampli- 
tude varying from 3-10 km per sec. The variations are characterized by a rapid 
deceleration. 



(2) There are at least three possible ways of accounting for the variability and 
reversal of stellar magnetic fields. The obvious explanation that the star possesses a 



V. C. A. Ferraro 331 

true variable magnetic field, in the sense that it is caused by alternating currents in 
the deep interior, has not been entertained seriously by astrophysicists on the 
grounds that t